Series and Convergence

Transcription

1 Chapter 5 Seres and Convergence We know a Taylor Seres for a functon s a polynomal approxmatons for that functon. Ths week, we wll see that wthn a gven range of x values the Taylor seres converges to the functon tself. In order to fully understand what that means we must understand the noton of a lmt, and convergence. 5. Lmts of functons and L hoptal s rule We begn wth a bref revew of lmts. You have probably studed lmts of functons before. Intutvely, lm f (x) = L means the closer x gets to a the closer the value f (x) gets to L. Indeed, as close as you want to get to L, say wthn.000 you can fnd an x such that f (x ) L.000. Some lmts are easy to fnd. For example, lm x = a and lm cf (x) = c lm f (x). If we happen to know that lm f (x) = L and lm g (x) = L then t s true that: () lm f (x) + g (x) = L + L () lm f (x)g (x) = L L () lm f (x)/g(x) = L /L,fL = 0. You may recall that t s more dffcult to fnd lm f (x)/g(x) when lm f (x) = 0, or and 5x 5 3x lm g (x) = 0, or. For example consder lm x 7x 5. Both the numerator and denomonator 3x go to,butthefractongoesto5/7. One way to see ths s to graph the functon! Here s another method. Theorem (L Hoptal s rule): If lm f (x)/g (x) s of ndetermnate form (0/0 or± / ± ) then Here are some examples. lm f (x)/g(x) = lm f (x)/g (x) lm x (3x + 4)/(7x 3) = lm x 6x/7 =. lm x (3x + 4x 0)/(x 3 8) = lm(6x + 4)/(3x ) = 4/3. x We can now make a general statement about lmts of quotents of polynomals. Theorem: If f (x) and g (x) are polynomals then:

2 Chapter 5: Seres and Convergence lm f (x)/g(x) = x f deg f (x) >g (x) 0 f deg f (x) <g (x) a k /b k f deg f (x) = g (x) where a k and b k are the coeffcents of the hghest terms n f (x) adn g (x) respectvely. 5. Lmts of sequences A sequence s a functon whose doman s all postve ntegers. For example,, 4, 8, 6,... s a sequence whose nth term s n. We wrte ths as the sequence a n = n. Lst the frst few terms of the sequence 3n, what s the 34th term of ths sequence? In ths class we wll mostly be nterested n lmts of sequences as x. It s not too hard to beleve that lm n =, and that lm = 0. Intutvely, we understand that as n gets really bg, then 3n 3n gets really close to 0. Now we try to formalze that dea. Defnton: lm a n = L f for any ɛ>0 there exsts an nteger N so that whenever n > N t s true that a n L <ɛ. We say the sequence a n converges to L n ths case. For example, lm = 0 snce, whenever n > /(3ɛ). I.e., norder to make 3n 3n wthn ɛ =.00 of 0 we need to pck n > 3(.00) = Seres A seres s an nfnte sum, lke ( ) = + ( ) + ( ) + ( ) 3 + ( ) 4 + Taylor Seres are examples of seres. In ths secton we address the followng queston. If we look at partal sums of the seres, do the answers we get approach some lmt. I.e., lm n ( ) =? In other words does the sequence of partal sums a n = n ( ) converge to some lmt? We say that a seres converges f such a lmt exsts and s fnte and dverges otherwse. Here are three examples, the frst seres converges and the second and thrd dverge. ( ) = ( ) = In order for a seres to converge t must be true that the terms n t get smaller and smaller. More exactly:

3 Chapter 5: Seres and Convergence 3 Theorem: If b converges, then lm b = 0. Ths condton s needed for a seres to converge but s not suffcent to nsure convergence! So f we have a seres b, and lm b = 0 then the seres must dverge. The second and thrd examples above are examples of ths. But t s possble that a seres a, has lm a = 0 and doesn t converge! A surprsng example s: The Geometrc Seres The frst seres we wll talk about s called the geometrc seres. It s of the form x = + x + x + x 3 + x 4 Notce that the very frst seres mentoned at the top of ths page s such a seres wth x =. Whether ths seres converges or not wll depend on what x s. We frst look at the smple case that x =. It s useful to defne the partal sums here and study there behavor. n ( ) S n = =? Calculatng these we see, S 0 =, S = 3, S = 7 4, S 3 = 5 8, S 4 = 3 6,.Weseeapattern!Indeed ( ) = lm ( n+ ) S n = lm n = lm For the general geometrc seres, we agan look at partal sums. n S n = x = + x + x + +x n ( n ) = In ths case, some algebra proves useful. Consder multplyng the partal sum by ( x). ( x)s n = ( x)( + x + x + x n ) = x n+ Ths means that S n = xn+ x So we need to determne the lmt lm S x n+ n = lm x Notce that t s a lmt as n goes to nfnty. The only n s n the numerator. Whether ths converges or not wll depend on what x s. In the example when x = the seres converges. When x = on the otherhand, the seres wll dverge. The general rule s that the seres wll converge as long as x <. Why? So f x < x = lm S x n+ n = lm x Check that ths corresponds wth what we got when x =. = x

4 Chapter 5: Seres and Convergence 4 Repeatng decmals You all know that = 3, but here s a proof of ths fact = = ( ) = Alternatng Seres ( ) ( ) = 3. = 3 = 9 3 An alternatng seres s one n whch the terms alternate n sgn, so t wll look lke ( ) n b n where b n wll be sequence. The followng theorem about alternatng seres wll be useful. Theorem: An alternatng seres ( ) b converges f and only f lm b = 0. For example, the seres ( ) converges, whle the seres ( ) n dverges. Compare ths result wth the prevous theorem. Ths one s f and only f, whle the prevous theorem was not. 5.6 Tests for Convergence and Absolute Convergence We contnue wth more ways of determnng whether a seres converges or not. Snce we already have a method whch determnes whether alternatng seres converge or dverge, ths week we wll concentrate on seres of postve terms Method : Comparson (I) If a and b are seres of postve terms and a b and the seres b converges, then so does the seres a. (II) If a and b are seres of postve terms and a b and the seres b dverges, then so does the seres a. Example: Show that the seres (+)! converges. We compare ths seres wth the seres, whch s a geometrc seres that converges to. We compare the th terms: ( + )! = 3 ( + ) ( ) = n= 5.6. Method : Integral Test If each a = f () for some contnuous decreasng functon f (x) then a converges f and only f f (x)dx converges. Example: Show that the seres dverges. The functon f (x) = x s contnuous and decreasng on the nterval (, ) and a =. dx = lm x m m dx = lm (ln m ln ) = x m

5 Chapter 5: Seres and Convergence 5 Example: The seres nterval (, ) and a =. p converges. The functon f p (x) = x s contnuous and decreasng on the p m dx = lm x p m Ths lmt s fnte f p > and nfnte otherwse. ( )( ) dx = lm x p m p m p Theorem: The seres p converges f p > and dverges f p. Absolute Convergence Sometmes a seres s nether all postve term nor ncely alternatng n sgn. An example s cos, some of whose terms are negatve, some postve but not every other one. In ths case we sometmes talk about the absolute convergence of a seres. The seres a s sad to converge absolutely f the seres of the absolute values of the terms a =a +a + converges. It s mportant because of the followng result: Theorem: If a seres converges absolutely then t converges. Example: Lookng at our example cos must converge snce wth ) Method 3: Rato Test Ths test s a generalzaton of the comparson test above. Ths tests for absolute convergence. Theorem: a + a converges f and only f lm <. a cos converges (by comparson Example: RecalltheGeometrcseress+ x + x + +c +. The rato test looks at the rato of the terms a + = x + and a = x : a + x + lm = lm a x =x. Ths lmt exsts and s < exactly when x <. Thus the geometrc seres converges when x < whch agrees wth what we had determned before. 5.7 Intervals of Convergence Ths brngs us to another defnton. Seres that contan a varable x, say, may converge for only some values of x, The values for whch the seres does converge are collectvely called the nterval of convergence for that seres. For example, the geometrc seres has nterval of convergence < x <. Returnng to the Taylor seres, we often want to know for what values of x does the Taylor Seres of the functon converge to the functon tself. Ths wll always turn out to be an nterval around the pont a where we centered the Taylor Seres. Examples:

6 Chapter 5: Seres and Convergence 6 () n= nn x n. The rato test gve us that a n+ (n + ) ( n + )x n+ (n + ) ( n + )x (n) ( n + )x lm = lm a n n n x n = lm n n > lm n n > lm nx whch s unless x = 0. So the seres converges only f x = 0. () n= ( )n x n n! The rato test shows: a n+ lm = lm a n ( ) n+ x n+ (n+)! ( ) n x n n! x = lm (n + ) = 0 <. So ths seres converges for all x. That s, the nterval of convergence s < x <. 5.8 Problems for Chapter 5 Exercse 5.. Fnd the lmt n each case. x 3 3x 4x + 3x 3x + x (a) lm (b) lm x 7x 3 x 6x 3 (c) lm 000 x x ln(x) x sn(x) (d) lm (e) lm x x x x (f) lm x 0 x Exercse 5.. Each of the followng sequences converge to 0. In each case fnd N (as a functon of ɛ) as the formal defnton of convergence requres. (a) a n = (b) a n n = (c) a 4n 3 n = (hnt: use logs) n Exercse 5.3. Decde f each of these geometrc seres converges and f so determne what t converges to. (a) (b) ( 5 ) 9 (c) ( 9 ) 5 Exercse Fnd the ratonal numbers represented by each of the repeatng decmals below. (a) (b) Exercse 5.5. Fnd a formula for the sum of each of the followng seres by performng sutable operatons on the geometrc seres. (a) x 3 + x 6 x 9 + (b) x + x 6 + x 0 + (c) x + 3x 4x 3 + (d) x + x + 3x 3 + 4x 4 + (e) x + x + x3 3 + x4 4 + Exercse 5.6. Whch of the followng alternatng sums converge and whch dverge? (a) ( ) (b) ( ) (d) ( ) 5 4 (c) ( ) ln (e) ( ) ln

7 Chapter 5: Seres and Convergence 7 Exercse 5.7. Usetheratotesttofnd the nterval of convergence for each of the followng power seres. x x (a) (b) (c) (d) x (g) ( + x) 99 (h) n n xn (e) n= x (f) Exercse 5.8. Fnd the nterval of convergence for the Taylor seres of the followng functons. () x! 3 n!x n (a) sn(x) (b) e x (c) cos(x) Exercse 5.9. Use the Comparson Test to determne whether each of these seres converges or dverges. (a) + (b) = ln n= (c) ( 9 5 Exercse 5.0. Use the ntegral test to determne f each of the followng seres converges or dverges. (a) ( (b) + ).3 (c) (d) n= n= n + (e) ln () (f) n= n= ) n 3 n n Exercse 5.. Use the rato test to fnd the nterval of convergence for each of the followng power seres. x n x n x n (a) (b) n n (c) n! (d) nx n n= (e) n= n= (g) ( + x) n 9 n (h) n 9 xn n= n n xn (f) Exercse 5.. Fnd the nterval of convergence for the Taylor seres of the followng functons. n= () n= n= n 3 x n n!x n (a) sn(x) (b) e x (c) cos(x) n=