ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series


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1 ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a where { } is a sequence of real numbers. Here formal means that we don t necessarily identify this expression with a particular numerical value, a sum. We will see shortly that sometimes such identification makes sense, in other cases not. A series is often abbreviated to. It is not straightforward to define the sum of a series. Naive approaches may lead to contradictions. Consider, for example, the series + ( ) + ( ) + + ( ) +. Grouping the terms in pairs 2, 3 4, 5 6, etc., we appear to get the sum zero for the series. On the other hand, grouping the terms 2 3, 4 5, 6 7, etc., we leave the first term outside groups, so the sum appears to be equal to. The standard definition of a sum is the result of the following two definitions. Definition.. The nth partial sum of the series is the sum of the first n terms, n S n := a i = a + a i= Definition.2. The series converges if the sequence {S n } of partial sums converges (equivalently, is a Cauchy sequence). The limit of this sequence, lim n S n, is a real number and is called the sum of the series. Example.3 (Geometric series). Consider k n = + k + k 2 + k 3 + where k > 0 and k. By a well known formula, S n = kn k.
2 2 ISSUED 24 APRIL 206 If k <, then lim n k n = 0 hence the limit of the numerator is, while the denominator is constant. Thus the sum of the series exists and is equal to lim S n = n k. If k >, then the numerator in S n tends to, while the denominator is still constant. Hence the sequence {S n } tends to infinity, and the series diverges. Example.4 (Harmonic series). This is the series n = Intuitively it is not at all clear whether it converges or not. Here is a medieval proof of the fact that harmonic series actually diverges. In the sum replace the part by a smaller sum the part , by a smaller sum and so on. As a result, we get a series, whose partial sums, starting from the third, are smaller than corresponding partial sums of the harmonic series. The new series is the same as and diverges, since its partial sums tends to infinity as n. Hence the partial sums of the larger harmonic series tend to infinity as n. Thus, the harmonic series diverges. Example.5 (Series of inverse squares). Consider the series n 2. This is a difficult one. Euler calculated that the sum of this series is π 2 /6. Example.6. The series converges for all integer k 2. It is easy to prove, knowing that the series on inverse squares converges, because the partial sums of the latter a greater than the corresponding partial sums of the given series. n k Leonhard Euler ( ) was a great Swiss mathematician. Worked mostly in Russian Imperial Academy of Sciences.
3 LECTURE NOTES 8 3 Example.7. The series ( ) n+ = n converges to ln 2. The proof is again nontrivial. In the next section we will show how to prove such and similar statements in general. We take a function (ln x in this case) and try to represent it as a series, called Taylor series. Note that this series resembles the harmonic series but the signs alternate in terms. This is called alternating harmonic series. 2. Operations on series We are not proving theorems listed in this section. The proofs can be found in standard textbooks, and will not be examined. There is a following relation between convergences of sequences { } and {S n }. Theorem 2.. If the series converges, then the sequence { } converges and lim n = 0. The converse statement to this theorem is not true, for example, the harmonic series diverges while lim n = 0. The converging series behave like vectors with respect to addition and multiplication by a constant. Theorem 2.2. If converge with sums a and b respectively, then for any α, β R the series (α + βb n )) converges with the sum αa + βb). and The following statement is a convenient tool for proving convergence, we already used it informally in Section. Theorem 2.3. Let { } and {b n } be two sequences such that 0 b n for each n, then if converges, then converges. b n b n
4 4 ISSUED 24 APRIL 206 Definition 3.. A series 3. Absolute convergence is absolutely converging if the series of absolute values is converging. One can prove that absolutely converging series converge (in the usual sense). The example of the alternating harmonic series shows that the converse is not true. Example 3.2. The series 2 n! = + x + x2 2! + + xn n! + converges absolutely for all x R. This is because for any x R there exists a sufficiently large n such that 0 n! < n 2, and now we can apply Theorem 2.3 to series n! and n 2. The sum of the series we denote by e x. Why this notation makes sense we will see in the next section. n! 4. Taylor series Let a function f : (a, b) R be infinitely many times differentiable on (a, b), i.e., there exist derivatives f (x 0 ), f (x 0 ),..., f (n) (x 0 ),... at every point x 0 (a, b). Fix an arbitrary x 0 (a, b) and consider the series f (n) n! (x x 0 ) n = f(x 0 ) + f (x 0 )! (x x 0 ) + f (x 0 ) (x x 0 ) 2 + 2! It is called Taylor series of f at x 0. In case x 0 = 0 the series is sometimes called Maclaurin series. It might happen that for each x (a, b) Taylor series of f at x 0 converges. There is a theorem that in this case its sum coincides with f on (a, b). Functions for which this takes place are called analytic. Note that there are infinitely differentiable functions which are not analytic (examples are nontrivial). 2 In the formula, n! = 2 n, and we assume, as it is customary, that 0! =.
5 LECTURE NOTES 8 5 Example 4.. The function e x is analytic on every interval ( a, a) R. Indeed, its Taylor series at x 0 = 0 can be explicitly computed: f(0) = e 0 =, f (0) = e 0 =,... and substituting in the general formula we get e x = + x + x2 2! + + xn n! + This series converges at every x R (see Example 3.2), hence it represents e x on the whole R. The sum of the series may be taken as a definition of e x. Example 4.2. The function sin x is also analytic on any interval ( a, a) R. Indeed, sin 0 = 0, sin 0 = cos 0 =, sin 0 = sin 0 = 0, sin 0 = cos 0 =, sin (4) 0 = sin 0 = 0. We see that in the sequence of higher derivatives at 0 the values repeat periodically. It follows that sin x = x x3 3! + x5 5! x7 7! + on R. Example 4.3. Consider the function with x. We have: f(x) = x f (0) = ( x) 2 (0) =, f 2 (0) = (0) = 2, ( x) 3 f (0) = 2 3 ( x) 4 (0) = 3!, f (4) = (0) = 4!,... ( x) 5 It is easy to see that f (n) (0) = n! It follows that the Taylor series for f(x) at x 0 = 0 is the geometric series, (see Example.3):, but this series does not converge for all x R. It converges for all x R such that x <. It follows that the function f(x) is analytic and representable by the Taylor series on the interval (, ) R. Example 4.4. It is straightforward to calculate that the function ln( + x) has a Taylor series ( ) n + = x x2 n x3 3 at x 0 = 0. The function is analytic and the series converges for all x R such that < x. In particular, for x = we recover the series from Example.7.
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