# Representation of functions as power series

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions as power series. Representation of Functions as Power Series (8.6). Theory In this section, we develop several techniques to help us represent a function as a power series. More precisely, given a function f (x), we will try to nd a power series c n (x a) n such that f (x) c n (x a) n. Part of the work will involve nding the values of x for which this is valid. Part of the reason for doing this is that a power series looks like a polynomial (except that it has in nitely many terms). Polynomials are among the easiest functions to work with. They are easy to di erentiate, integrate,... So, if f is a complicated function, replacing it with a power series amounts to replacing it with a polynomial. Therefore, working with f becomes easier. There are some technical di culties to resolve, we will address those as we develop the technique. First, we will look at techniques which will allow us to obtain a series representation by using known series representations. The techniques involved are substitution, di erentiation and integration. Then, we will learn a technique which will allow us to nd a series representation for a given function f directly, without having to use known series. We now look at each technique (substitution, di erentiation and integration) in details, using examples. At this point, we only know the following series representation: x + x + x + + ::: in ( ; ) x n

2 When nding the power series of a function, you must nd both the series representation and when this representation is valid (its domain).. Substitution We derive the series for a given function using another function for which we already have a power series representation. Then, we do the following:. Figure out which substitution can be applied to transform the function for which we know the series representation to the function for which we want a series representation.. Apply the same substitution to the known series. This will give us the series representation we wanted. 3. The domain of the new function is obtained by applying the same substitution to the domain of the known series. Example Find a power series representation for f (x) and its domain. + x We use the representation of, replacing x by x. We obtain: x + x ( x) + ( x) + ( x) + ( x) 3 + ::: x + x + ::: ( ) n x n The series representation for was valid if jxj <. If we apply the same x substitution, we see that this representation is valid if j xj < or jxj <. In other words, the interval of convergence is also ( ; ). Example Find a power series representation for f (x) x. Proceeding as above and remembering that x X x n, we have: x x n x n + x + x 4 + :::

3 This is a geometric series which converges when x < that is when x < or < x <. Example 3 Find a power series representation for and nd its domain. + x First, we rewrite + x + x. Now, we nd a power series representation for + x, then we will multiply it by. + x can be obtained from x by x replacing x by. Since x X x n, we have: Therefore, x n + x ( ) n x n + x ( ) n x n ( ) n x n n+ x converges when jxj <, so this series converges when x < ) jxj <. So, the interval of convergence is ( ; ), the radius of convergence is. Remark 4 In the rst of these two examples, we applied the substitution to the expanded form of. In the second, we applied it to the compact form. You x can do it either way, it is simply a matter of choice. Example 5 Suppose that you are given that a series repersentation for e x is e x x n in ( ; ). What is a series representation for e x and what is its domain? We go from e x to e x using the substitution x! x. Thus e x x n x n also valid in ( ; ) 3

4 .3 Di erentiation and Integration The driving force behind the integration and di erentiation techniques is the theorem below which we state without proof. Theorem 6 If the power series c n (x R > 0 then the function de ned by f (x) a) n has a radius of convergence c n (x a) n c 0 + c (x a) + c (x a) + ::: is di erentiable (hence) continuous on (a R; a + R) and. f 0 (x) c + c (x a) + 3c 3 (x a) + ::: In other words, the series can be di erentiated term by term.. R (x a) (x a) 3 f (x) dx C + c 0 (x a) + c + c + ::: In other words, 3 the series can be integrated term by term. 3. Note that whether we di erentiate or integrate, the radius of convergence is preserved. However, convergence at the endpoints must be investigated every time. Remark 7 This theorem simply says that the sum rule for derivatives and integrals also applies to power series. Remember that a power series is a sum, but it is an in nite sums. So, in general, the results we know for nite sums do not apply to in nite sums. The theorem above says that it does in the case of in nite series. Remark 8 The formula in part of the theorem is obtained simply by di erentiating the series term by term. Since f (x) c 0 + c (x a) + c (x a) + ::: then 0 f 0 (x) c 0 + c (x a) + c (x a) + ::: (c 0 ) 0 + (c (x a)) 0 + c (x a) 0 + ::: 0 + c + c (x a) + 3c 3 (x a) + ::: () Alternatively, one can also di erentiate the general formula. Since f (x) c n (x a) n 4

5 then f 0 (x)! 0 c n (x a) n (c n (x a) n ) 0 by the theorem nc n (x a) n The rst term of this series (when n 0) is 0, thus we can start summation at n. Hence, we have f 0 (x) nc n (x a) n () n The reader should check that formulas and are identical. Remark 9 The formula in part of the theorem is obtained by integrating term by term. It can also be obtained by integrating the general formula. Since f (x) c n (x a) n then Z f (x) dx Z X c n (x a) n! dx Z (c n (x a) n ) dx by the theorem Since an antiderivative of c n (x a) n is C + c n (x a) n+, we have n + Z If we expand this, we get Z f (x) dx C + c n (x a) n+ n + f (x) dx C + c 0 (x a) + c (x a) Which is the formula which appears in the theorem. + c (x a) ::: 5

6 Example 0 Given that a power series representation for f (x) is f (x) + x + x + + ::: nd a power series representation for f 0 (x) and R f (x) dx. x n First, we nd f 0 (x). From the theorem, we know it is enough to di erentiate term by term. Thus, f 0 (x) x + 3x + ::: + x + 3x + ::: Note that we can also obtain the same result by using the general formula. In this case, f (x) x n Thus f 0 (x) Which gives us the same answer. (x n ) 0 n nx n nx n Next, we nd R f (x) dx. From the theorem, it is enough to integrate term by term. Thus since f (x) + x + x + + :::, we have: Z f (x) dx C + x + x + x3 3 + x4 4 + ::: Alternatively, we can work from the general formula. Since f (x) we have Which is the same formula. Z f (x) dx Z (x n ) dx C + x n+ n + x n, 6

7 .3. Di erentiation This time, we nd the series representation of a given series by di erentiating the power series of a known function. More precisely, if f 0 (x) g (x), and if we have a series representation for f and need one for g, we simply di erentiate the series representation of f. The theorem above tells us that the radius of convergence will be the same. However, we will have to check the endpoints. Example Find a series representation for convergence. We begin by noting that we have:, nd the interval of ( x) 0 x ( x). Since x +x+x + +::::, ( x) + x + x + + ::: 0 + x + 3x :::: nx n n The radius of convergence is still. Since converges for x in ( ; ), we x know that will also converge in ( ; ). It might also converge at the ( x) endpoints, so we need to check them. Do it as an exercise. Remark If you prefer to work from the compact form of the series, it can be done. x X x n ( x)! 0 x n (x n ) 0 n nx n However, you have to be careful with the starting value of n. Example 3 Suppose you know that a series representation for sin x is sin x x 3! + x5 5! x 7 7! + ::: ( ) n x n+ (n + )! 7

8 Find a power series representation for cos x. cos x (sin x) 0 x 3! + x5 5! x! + x4 4! ( ) n x n (n)! x 7 0 7! + ::: x 6 6! + ::: Example 4 Find a power series representation for xe x repersentation for e x is e x x n in ( ; ). We can do this problem two ways. given that a series Method Using substitution, we can nd a power series representation for e x, then multiply what we nd by x. The rst part was done earlier, x n and we found that e x on ( ; ). Thus, xe x xx n x n+ also in ( ; ). Method We note that xe x e x 0. So, using substitution, we can nd a power series representation for e x, then di erentiate it to get a series 8

9 representation for xe x. xe x e x 0 n n! 0 x n nx n nx n x n (n )! x n+ when n 0, nxn 0 So, either way, we nd the same answer..3. Integration This time we nd the power series representation of a function by integrating the power series representation of a known function. If g (x) R f (x) dx and we know a power series representation for f (x), we can get a series representation for g (x) by integrating the series representation of f. Example 5 Find a power series representation for ln ( x). We know that ln ( x) R dx. By our earlier work, we found that x x + x + x + + ::: in ( ; ), so ln ( x) C + x + x + x3 3 + ::: We also know that ln 0, From the above equation, we get that C 0 by letting x 0. Therefore ln ( x) x + x + x3 3 + ::: x n+ n + Therefore, ln ( x) x n+ n + 9

10 Since the original series converges in ( ; ), this one will also converge there. The end points should be checked, we leave it as an exercise. Example 6 Find a power series rpresentation for tan x. We use the fact that tan x R se see that dx + x. First, since x +x+x + +:::, + x x + x 4 x 6 + ::: and therefore tan x C + x 3 + x5 5 Using the fact that tan 0 0 gives us C 0. Thus tan x x.4 Things to Know 3 + x5 5 x ::: x ::: Be able to nd the series representation of a function by substitution, integration or di erentiation. Problems assigned: #, 3, 5, 7,, 3,, 35 on pages 604, 605 Taylor and Maclaurin s Series (8.7). Introduction The previous section showed us how to nd the series representation of some functions by using the series representation of known functions. The methods we studied are limited since they require us to relate the function to which we want a series representation with one for which we already know a series representation. In this section, we develop a more direct approach. When dealing with functions and their power series representation, there are three fundamental questions one has to answer:. Does a given function have a power series representation?. If it does, how do we nd it? 3. What is the domain in other words, for which values of x is the representation valid? Question is more theoretical, we won t address it. We will concentrate on questions and 3. Assuming f has a power series representation that is f (x) c 0 +c (x a)+ c (x a) + :::, we want to nd what the power series representation is, that is we need to nd the coe cients c 0 ; c ; c ; :::. It turns out that it is not very di cult. The technique used is worth remembering. We rst nd c 0. Having found c 0 we next nd c. Then, we nd c and so on. 0

11 Finding c 0. Since f (x) c 0 + c (x a) + c (x a) + c 3 (x a) 3 + c 4 (x a) 4 + ::: (3) it follows that f (a) c 0 + c (0) + c (0) + ::: c 0 Finding c. Di erentiating equation 3 gives us f 0 (x) c + c (x a) + 3c 3 (x a) + 4c 4 (x a) 3 + ::: Therefore, f 0 (a) c Finding c. We proceed the same way. First, we compute f 00 (x), then f 00 (a). Therefore or f 00 (x) c + () (3) c 3 (x a) + (3) (4) c 4 (x a) + ::: f 00 (a) c c f 00 (a) In general. Continuing this way, we can see that. De nitions and Theorems c n f (n) (a) Theorem 7 If the function f has a power series representation, that is if f (x) c n (x a) n c 0 + c (x a) + c (x a) + ::: for jx aj < R then its coe cients are given by: In other words c n f (n) (a) f (x) f (a) + f 0 (a) (x a) + f 00 (a) (x a) De nition 8. The series in the previous theorem is called the Taylor series of the function f at a. + f 000 (a) (x a)3 3! f (n) (a) + ::: (x a) n!

12 . The nth partial sum is called the nth order Taylor polynomial It is denoted T n : So, nx f (i) (a) T n (x) (x a) i i! i0 3. In the special case a 0, the series is called a Maclaurin s Series. So, a f (n) (0) Maclaurin s series is of the form x n and the Maclaurin s series for a function f is given by f (x).3 Examples f (n) (0) x n f (0) + f 0 (0) x + f 00 (0) x + f 000 (0) x3 3! + ::: We now look how to nd the Taylor and Maclaurin s series of some functions. Example 9 Find the Maclaurin s series for f (x) e x, nd its domain. f (n) (0) The series will be of the form x n, we simply need to nd the coef cients f (n) (0). This is easy. Since all the derivatives of e x are e x, it follows that f (n (x) e x, thus f (n) (0) e 0, hence e x f (n) (0) x n x n To nd where this series converges, we use the ration test and compute: lim a n+ a n lim Thus the domain is all real numbers. x n+ (n + )! x n jxj n+ lim (n + )! jxj n jxj lim n + 0

13 Example 0 Find a Taylor series for f (x) e x centered at. f (n) () The series will be of the form (x ) n, we simply need to nd the coe cients f (n) (). This is easy. Since all the derivatives of e x are e x, it follows that f (n (x) e x, thus f (n) () e, hence e x f (n) () e (x )n (x ) n Example Find the n th order Taylor polynomial for e x centered at 0 and centered at. Plot these polynomials for n ; ; 3; 4; 5. What do you notice? We already computed the power series corresponding to these two situations. We found that and e x + x + x! + x3 3! + x4 4! + x5 5! + x6 6! + ::: e x e + e (x ) (x )3 (x ) + e + e + :::! 3! e + (x ) + (x )! + (x )3 3! + (x )4 4! + (x )5 5! + :::! If we denote T n the n th order Taylor polynomial for e x centered at 0 and Q n the n th order Taylor polynomial for e x centered at, we have: n th order Taylor polynomials for e x centered at 0. T (x) + x T (x) + x + x! T 3 (x) + x + x! + x3 3! T 4 (x) + x + x! + x3 3! + x4 4! T 5 (x) + x + x! + x3 3! + x4 4! + x5 5! The graphs is shown on the next page. 3

14 Graphs of e x and the rst 5 Taylor polynomials centered at 0. The functions have the following colors: e x : black T : blue T : red T 3 : green T 4 : purple T 5 : yellow 4

15 n th order Taylor polynomial for e x centered at : Q (x) e ( + (x )) Q (x) e + (x ) + Q 3 (x) e + (x ) + Q 4 (x) e + (x ) + Q 5 (x) e + (x ) +! (x )! (x )! (x )! (x )! The graphs are shown on the next page ! (x )3 3! (x )3 3! (x )3 3! + +! (x )4 4! (x )4 4! +! (x )5 5! 5

16 Graphs of e x and the rst 5 Taylor polynomials centered at. The functions have the following colors: e x : black Q : blue Q : red Q 3 : green Q 4 : purple Q 5 : yellow We can see the following: The Taylor polynomial approximates the functions well near the point at which the series is centered. As we move away from this point, the approximation deteriorates very quickly. The approximation is better the higher the degree of the Taylor polynomial. More speci cally, the Taylor polynomial stay closer to the function over a larger interval, the higher its degree. 6

17 Example Find a Maclaurin s series for f (x) sin x and nd its domain f (n) (0) The series will be of the form x n, we simply need to nd the coe - cients f (n) (0). That is we need to nd all the derivatives of sin x and evaluate them at x 0. We summarize our ndings in the table below: n f (n) (x) f (n) (0) 0 sin x 0 cos x sin x 0 3 cos x 4 sin x 0 So, we see that we will have coe cients only for odd values of n. In addition, the sign of the coe cients will alternate. Thus, since we have f (x) f (0) + f 0 (0) x + f 00 (0) x + f 000 (0) x3 3! + f (4) (0) x4 4! + ::: sin x 0 + x + 0 x 3! + x5 5! 3! x5 5! x 7 7! + ::: ( ) n x n+ (n + )! To nd where this series converges, we use the ration test and compute lim a n+ a n. Since a n xn+ (n + )!, a n+ xn+3 (n + 3)!. Therefore, x n+3 lim a n+ a n lim (n + 3)! x n+ (n + )! jxj n+3 (n + )! lim jxj n+ (n + 3)! jxj lim (n + )! (n + 3)! ::: jxj lim (n + ) (n + 3) 0 Thus the series representation is valid for all x. 7

18 Here are some important functions and their corresponding Maclaurin s series x + x + x + + x 4 + ::: ( ; ) e x + x + x! + x3 + ::: ( ; ) 3! sin x x cos x tan x x 3! + x5 5! x! + x4 4! 3 + x5 5 x 7 7! x 6 + ::: ( ; ) + ::: 6! ( ; ) x 7 + ::: 7 [ ; ] Remark 3 The n th order Taylor polynomial (T n ) associated with the series expansion of a function f is the n th degree polynomial obtained by truncating the series expansion and keeping only the terms of degree less than or equal to n. In the case of sin x, since It follows that and so on. sin x x 3! + x5 5! T x T x T 3 x T 4 x T 5 x T 6 x 3! 3! 3! + x5 5! 3! + x5 5! x 7 7! + :::.4 Summary We now have four di erent techniques to nd a series representation for a function. These techniques are:. Substitution. Di erentiation 3. Integration 8

19 4. Taylor/Maclaurin s series. It may appear that the last technique is much more powerful, as it gives us a direct way to derive the series representation. In contrast, the rst three techniques require we start from a known series representation. However, the rst three techniques should not be ignored. In many cases, they make the work easier. We illustrate this with a few examples. Example 4 Find a Maclaurin s series for f (x) e x. Of course, this can be done directly. But consider the problem of nding all the derivative of e x. One can also start from e x and substitute x for x. Since it follows that e x x n + x + x + x3 3! + x4 4! e x x n ( ) n x n This was pretty painless. To convince yourself that this is the way to do it, try the direct approach instead! Example 5 Find a Maclaurin s series for cos x. Again, one can try the direct approach as in example. It is not too di cult. One can also realize that (sin x) 0 cos x. Thus, one can start with the series for sin x (which we already derived) and nd the series for cos x by di erentiating it. We get cos x (sin x) 0 ( ) n x n+ (n + )! ( ) n x n+ ( ) n x n (n)! x + x4 4! (n + )! x 6 6!! 0 0 9

20 .5 Things to Remember Given a function, be able to nd its Taylor or Maclaurin s series. Be able to nd the radius and interval of convergence of Taylor or Maclaurin s series. Section 8.7: # 3, 5, 7, 9,, 3, 9,, 3, 5, 3 on pages 65, Applications The purpose of this section is to show the reader how Taylor series can be used to approximate functions. The approximation can then be used to either evaluate a function at speci c values of x, to integrate or to di erentiate the function. Of course, we would only use this technique with functions for which the traditional calculus methods do not work. For example, we may need to compute Z 0 e x dx. This cannot be done using the integration techniques learned in a traditional calculus class since e x does not have an antiderivative which can be expressed in terms of elementary functions. Another application might be to approximate sin (0:0), without a calculator. The di culty does not lie in the series representation of a given function, we now know how to represent functions as power series. However, series have in nitely many terms, for practical purposes, we can only use a nite number of them. Thus, we replace the in - nite series by the corresponding Taylor polynomial (see de nition 8) of order n (T n ), for some n. We then use the Taylor polynomial instead of the function. This can be used to evaluate a function, integrate a function or di erentiate a function. However, when we perform the following approximation f (x) f (a) + f 0 (a) (x a) + f 00 (a)! (x a) + ::: + f (n) (a) there are several questions to answer before we can carry it out: (x a) n. How do we pick a, the number around which the series is centered?. How do we pick n so the Taylor polynomial T n approximates the given function f within the desired accuracy? Answer to question There are several factors to take under consideration. First, since we have to evaluate f (n) (a), a must be picked so we can do this evaluation easily. Second, you will recall that the accuracy of the approximation decreases as x gets further away from a. Therefore, we need to pick a so that the values at which f (x) will be approximated are in the domain of the series, and not too far from a. For example, if we had to approximate sin (:00), then a 0 would be a good choice because it satis es both criteria. 0

21 Answer to question Once we have selected a and have a series representation for f, we use the techniques studied to approximate a series and nd the error. The examples below illustrate these applications. Example 6 Find the nth order Taylor polynomial for f (x) cos x when n ; 3; 4; 5; 6. Sketch the graph of cos x as well as the Taylor polynomials found. We already know the power series for cos x. So, cos x P (x) P 3 (x) P 4 (x) P 5 (x) P 6 (x) x! + x4 4! x! x! x! + x4 4! x! + x4 4! x! + x4 4! x 6 6! + x8 8! :::: You will note that because every other coe cient in the series expansion of cos x is 0; P 3 P, P 5 P 4. The graph of these polynomials is shown on gure. Remark 7 You will notice that as n increases, P n gets closer to the graph of cos x. In other words, the accuracy of our approximation increases with n. However, P n is a good approximation for cos x in a neighborhood of 0, as we move away from 0, P n gets further and further away from cos x. This is important. When one approximates a function with a Taylor polynomial about a, the approximation is good only for values of x close to a. Example 8 Approximate cos 0:0 with an error less than 0 0. First, we note that since :0 is close to 0, we can use a Taylor polynomial centered at 0 to approximate cos (:0). Therefore, using the Taylor polynomial for cos x centered at 0, and replacing x by 0:0, we get: cos 0:0 nx i0 ( ) i 0:0 i (i)! We need to nd n so that if we approximate ( ) i 0:0 i nx by ( ) i 0:0 i (i)! (i)!, i0 i0 the error is less than 0 0. We notice that ( ) i 0:0 i is an alternating (i)! i0 x 6 6!

22 Figure : Graph of cos x and some Taylor polynomials series with b n 0:0n, so we know how to estimate its error. The error is (n)! always less than b n+. So, if we want the error to be less than 0 0, it is enough to solve: 0:0 (n+) (n + )! b n+ < 0 0 < 0 0 We solve this by trying various values of n. The table below shows this procedure:

23 0:0 (n+) n (n + )! We see that n 3 is enough. Therefore: cos 0:0 3X ( ) n 0:0 n (n)! 0: Example 9 Approximate R 0 e x dx with an error less than 0:00. First, we nd a series representation for e x. Since e x x n it follows that Therefore and therefore Z 0 Z e x e x dx e x dx ( ) n x n ( ) n x n+ (n + ) ( ) n x n+ (n + ) 0 ( ) n (n + ) This is an alternating series ( ) n b n with b n. From our (n + ) knowledge of alternating series, we know that if we approximate ( ) i (i + ) i! i0 nx by ( ) i (i + ) i!, the error will be less than b n+ (n + 3) (n + )!. So, i0 we nd n such that (n + 3) (n + )! < :00 3

24 We try several values of n, we nd that when n 3, and when n 4, : So, (n + 3) (n + )! 4X gives us the desired approximation. ( ) n 0: (n + ) (n + 3) (n + )! : Problems The problems assigned for power series, Taylor series and representation of functions as power series were: Section 8.6: #, 3, 5, 7,, 3,, 35 on pages 60, 6. Section 8.7: # 3, 5, 7, 9,, 3, 9,, 3, 5, 3 on pages 65, 66. In addition, for the applications discussed in this section, do # 3, 5, 7 on page 68. 4

### 6.8 Taylor and Maclaurin s Series

6.8. TAYLOR AND MACLAURIN S SERIES 357 6.8 Taylor and Maclaurin s Series 6.8.1 Introduction The previous section showed us how to find the series representation of some functions by using the series representation

### Partial Fractions Decomposition

Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational

### 36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Manipulating Power Series

LECTURE 24 Manipulating Power Series Our technique for solving di erential equations by power series will essentially be to substitute a generic power series expression y(x) a n (x x o ) n into a di erential

### MATH Area Between Curves

MATH - Area Between Curves Philippe Laval September, 8 Abstract This handout discusses techniques used to nd the area of regions which lie between two curves. Area Between Curves. Theor Given two functions

### Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

### 4.6 Null Space, Column Space, Row Space

NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

### Sample Problems. Practice Problems

Lecture Notes Partial Fractions page Sample Problems Compute each of the following integrals.. x dx. x + x (x + ) (x ) (x ) dx 8. x x dx... x (x + ) (x + ) dx x + x x dx x + x x + 6x x dx + x 6. 7. x (x

### 1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

### Power Series. We already know how to express one function as a series. Take a look at following equation: 1 1 r = r n. n=0

Math -0 Calc Power, Taylor, and Maclaurin Series Survival Guide One of the harder concepts that we have to become comfortable with during this semester is that of sequences and series We are working with

### 160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

### Lectures 5-6: Taylor Series

Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

### Fourier Series Chapter 3 of Coleman

Fourier Series Chapter 3 of Coleman Dr. Doreen De eon Math 18, Spring 14 1 Introduction Section 3.1 of Coleman The Fourier series takes its name from Joseph Fourier (1768-183), who made important contributions

### A power series about x = a is the series of the form

POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

### Fourier Series Expansion

Fourier Series Expansion Deepesh K P There are many types of series expansions for functions. The Maclaurin series, Taylor series, Laurent series are some such expansions. But these expansions become valid

### Sequences and Series

Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite

### MATH 132: CALCULUS II SYLLABUS

MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early

### Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)

ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process

### c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.

Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions

### Notes on Chapter 1, Section 2 Arithmetic and Divisibility

Notes on Chapter 1, Section 2 Arithmetic and Divisibility August 16, 2006 1 Arithmetic Properties of the Integers Recall that the set of integers is the set Z = f0; 1; 1; 2; 2; 3; 3; : : :g. The integers

### Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

### Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by

Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes

### Partial Derivatives. @x f (x; y) = @ x f (x; y) @x x2 y + @ @x y2 and then we evaluate the derivative as if y is a constant.

Partial Derivatives Partial Derivatives Just as derivatives can be used to eplore the properties of functions of 1 variable, so also derivatives can be used to eplore functions of 2 variables. In this

### Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 28 Fourier Series (Contd.) Welcome back to the lecture on Fourier

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

### Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by

MATH 42, Fall 29 Examples from Section, Tue, 27 Oct 29 1 The First Hour Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x

### APPLICATIONS OF DIFFERENTIATION

4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

### Using Series to Solve Differential Equations

Using Series to Solve Differential Equations Many differential equations can t be solved explicitly in terms of finite combinations of simple familiar functions. This is true even for a simple-looking

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### MATH 2300 review problems for Exam 3 ANSWERS

MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

### LIES MY CALCULATOR AND COMPUTER TOLD ME

LIES MY CALCULATOR AND COMPUTER TOLD ME See Section Appendix.4 G for a discussion of graphing calculators and computers with graphing software. A wide variety of pocket-size calculating devices are currently

### Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

### Lecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if

Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close

### 1 Another method of estimation: least squares

1 Another method of estimation: least squares erm: -estim.tex, Dec8, 009: 6 p.m. (draft - typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i

Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions

### Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

### Section 3.1 Calculus of Vector-Functions

Section 3.1 Calculus of Vector-Functions De nition. A vector-valued function is a rule that assigns a vector to each member in a subset of R 1. In other words, a vector-valued function is an ordered triple

### 5 Indefinite integral

5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

### Taylor Series and Asymptotic Expansions

Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.

### Estimating the Average Value of a Function

Estimating the Average Value of a Function Problem: Determine the average value of the function f(x) over the interval [a, b]. Strategy: Choose sample points a = x 0 < x 1 < x 2 < < x n 1 < x n = b and

### Name: ID: Discussion Section:

Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 hand-graded questions worth a total of 30 points. INSTRUCTIONS:

### Section 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a

### 106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM 5.1.1 Fermat s Theorem f is differentiable at a, then f (a) = 0.

5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

### MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

### Trigonometric Identities

Trigonometric Identities Dr. Philippe B. Laval Kennesaw STate University April 0, 005 Abstract This handout dpresents some of the most useful trigonometric identities. It also explains how to derive new

### Review for Calculus Rational Functions, Logarithms & Exponentials

Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

### 2.2 Derivative as a Function

2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x

### 3 Contour integrals and Cauchy s Theorem

3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of

### 1 Error in Euler s Method

1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of

### Math 201 Lecture 23: Power Series Method for Equations with Polynomial

Math 201 Lecture 23: Power Series Method for Equations with Polynomial Coefficients Mar. 07, 2012 Many examples here are taken from the textbook. The first number in () refers to the problem number in

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### 10.2 Series and Convergence

10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

### 14.451 Lecture Notes 10

14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2

### Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

### Recap on Fourier series

Civil Engineering Mathematics Autumn 11 J. Mestel, M. Ottobre, A. Walton Recap on Fourier series A function f(x is called -periodic if f(x = f(x + for all x. A continuous periodic function can be represented

### Intrinsic Geometry. 2 + g 22. du dt. 2 du. Thus, very short distances ds on the surface can be approximated by

Intrinsic Geometry The Fundamental Form of a Surface Properties of a curve or surface which depend on the coordinate space that curve or surface is embedded in are called extrinsic properties of the curve.

### Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

### MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity

MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x

### Fourier Series. 1. Full-range Fourier Series. ) + b n sin L. [ a n cos L )

Fourier Series These summary notes should be used in conjunction with, and should not be a replacement for, your lecture notes. You should be familiar with the following definitions. A function f is periodic

### Lecture 1: Systems of Linear Equations

MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

### 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

### Section 2.4: Equations of Lines and Planes

Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y

### Analysis MA131. University of Warwick. Term

Analysis MA131 University of Warwick Term 1 01 13 September 8, 01 Contents 1 Inequalities 5 1.1 What are Inequalities?........................ 5 1. Using Graphs............................. 6 1.3 Case

### Welcome to Analysis 1

Welcome to Analysis 1 Each week you will get one workbook with assignments to complete. Typically you will be able to get most of it done in class and will finish it off by yourselves. In week one there

### 4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### 5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0

a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a

### Development Economics Lecture 5: Productivity & Technology

Development Economics Lecture 5: Productivity & Technology Måns Söderbom University of Gothenburg mans.soderbom@economics.gu.se www.soderbom.net 1 Measuring Productivity Reference: Chapter 7 in Weil. Thus

### 88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### Review Sheet for Third Midterm Mathematics 1300, Calculus 1

Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,

### Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =

Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a

### 4.3 Lagrange Approximation

206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Fourier Series, Integrals, and Transforms

Chap. Sec.. Fourier Series, Integrals, and Transforms Fourier Series Content: Fourier series (5) and their coefficients (6) Calculation of Fourier coefficients by integration (Example ) Reason hy (6) gives

### Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

### Linear and quadratic Taylor polynomials for functions of several variables.

ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Stirling s formula, n-spheres and the Gamma Function

Stirling s formula, n-spheres and the Gamma Function We start by noticing that and hence x n e x dx lim a 1 ( 1 n n a n n! e ax dx lim a 1 ( 1 n n a n a 1 x n e x dx (1 Let us make a remark in passing.

### Maths 361 Fourier Series Notes 2

Today s topics: Even and odd functions Real trigonometric Fourier series Section 1. : Odd and even functions Consider a function f : [, ] R. Maths 361 Fourier Series Notes f is odd if f( x) = f(x) for

### An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate.

Chapter 10 Series and Approximations An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate 1 0 e x2 dx, you could set

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### Chapter 11 - Curve Sketching. Lecture 17. MATH10070 - Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson.

Lecture 17 MATH10070 - Introduction to Calculus maths.ucd.ie/modules/math10070 Kevin Hutchinson 28th October 2010 Z Chain Rule (I): If y = f (u) and u = g(x) dy dx = dy du du dx Z Chain rule (II): d dx

### Differentiation and Integration

This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

### G. GRAPHING FUNCTIONS

G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### Integration Involving Trigonometric Functions and Trigonometric Substitution

Integration Involving Trigonometric Functions and Trigonometric Substitution Dr. Philippe B. Laval Kennesaw State University September 7, 005 Abstract This handout describes techniques of integration involving