Chapter 12: Infinite Series Section 12.1 Sigma Notation The Symbol Σ Section 12.2 Infinite Series Partial Sums Definition Example Geometric Series Theorem 12.2.4 Theorems 12.2.5 Section 12.3 The Integral Test; Basic Comparison, Limit Comparison Theorem 12.3.1 The Integral Test Harmonic Series and p-Series Properties of Convergence and Divergence Basic Comparison Theorem Applying the Basic Comparison Theorem Limit Comparison Theorem Applying the Limit Comparison Theorem   Section 12.4 The Root Test; The Ratio Test The Root Test Applying the Root Test The Ratio Test Applying the Ratio Test Summary on Convergence Tests Section 12.5 Absolute Convergence and Conditional Convergence; Alternating Series Absolute Convergence and Conditional Convergence Alternating Series Estimating the Sum of an Alternating Series   Section 12.6 Taylor Polynomials in x; Taylor Series in x Taylor’s Theorem Corollary: Lagrange Formula for the Remainder Taylor Series Properties Section 12.7 Taylor Polynomials and Taylor Series in x – a a. Taylor’s Theorem nth Taylor Polynomial for g in powers of x – a Taylor Expansion of g(x) in powers of x – a Section 12.8 Power Series Definition and Theorem Case 1, 2 and 3 Radius of Convergence Section 12.9 Differentiation & Integration of Power Series Theorem 12.9.1 The Differentiability Theorem Differentiating a Power Series Term-by-Term Integration Abel’s Theorem Power Series: Taylor Series Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Sigma Notation The symbol Σ is the capital Greek letter “sigma.” We write (1) (“the sum of the ak from k equals 0 to k equals n”) to indicate the sum a0 + a1 + · · · + an. More generally, for n ≥ m, we write (2) to indicate the sum am + am+1 + · · · + an. In (1) and (2) the letter “k” is being used as a “dummy” variable; namely, it can be replaced by any other letter not already engaged. For instance, all mean the same thing: a3 + a4 + a5 + a6 + a7. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Infinite Series Introduction; Definition To form an infinite series, we begin with an infinite sequence of real numbers: a0, a1, a2, . . . . We can’t form the sum of all the ak (there are an infinite number of them), but we can form the partial sums: Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Infinite Series Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Infinite Series Example The series illustrate two forms of divergence: bounded divergence, unbounded divergence. For the first series, sn = 1 − 1 + 1 − 1+ · · · +(−1)n. Here The sequence of partial sums reduces to 1, 0, 1, 0, . . . . Since the sequence diverges, the series diverges. This is an example of bounded divergence. For the second series, Since sn > 2n, the sum tends to ∞, and the series diverges. This is an example of unbounded divergence. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Infinite Series The Geometric Series The geometric progression 1, x, x2, x3, . . . gives rise to the numbers 1, 1 + x, 1 + x + x2, 1 + x + x2 + x3, . . . . These numbers are the partial sums of what is called the geometric series: This series is so important that we will give it special attention. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Infinite Series Setting x = ½ in (12.2.2), we have By beginning the summation at k = 1 instead of at k = 0, we drop the term 1/20 = 1 and obtain The partial sums of this series are given below and illustrated in Figure 12.2.1. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Infinite Series Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Infinite Series Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Integral Test; Basic Comparison, Limit Comparison Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Integral Test; Basic Comparison, Limit Comparison Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Integral Test; Basic Comparison, Limit Comparison Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Integral Test; Basic Comparison, Limit Comparison In the absence of detailed indexing, we cannot know definite limits, but we can be sure of the following: Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Integral Test; Basic Comparison, Limit Comparison Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Integral Test; Basic Comparison, Limit Comparison Applying the Basic Comparison Theorem Example (a) converges by comparison with (b) converges by comparison with Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Integral Test; Basic Comparison, Limit Comparison Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Integral Test; Basic Comparison, Limit Comparison Applying the Limit Comparison Theorem Example Determine whether the series converges or diverges. Solution For large k differs little from As k →∞, Since converges, (it is a convergent geometric series), the original series converges. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Root Test; The Ratio Test Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Root Test; The Ratio Test Applying the Root Test Example For the series The series converges. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Root Test; The Ratio Test Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Root Test; The Ratio Test Applying the Ratio Test Example For the series The series converges. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Root Test; The Ratio Test Summary on Convergence Tests In general, the root test is used only if powers are involved. The ratio test is particularly effective with factorials and with combinations of powers and factorials. If the terms are rational functions of k, the ratio test is inconclusive and the root test is difficult to apply. Series with rational terms are most easily handled by limit comparison with a p-series, a series of the form Σ 1/kp. If the terms have the configuration of a derivative, you may be able to apply the integral test. Finally, keep in mind that, if ak 0, then there is no reason to try any convergence test; the series diverges. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Absolute and Conditional Convergence Absolute Convergence and Conditional Convergence Series Σak for which Σ|ak| converges are called absolutely convergent. The theorem we have just proved says that Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Absolute and Conditional Convergence Alternating Series A series such as in which consecutive terms have opposite signs is a called an alternating series. As in our example, we shall follow custom and begin all alternating series with a positive term. In general, then, an alternating series will look like this: with all the ak positive. In this setup the partial sums of even index end with a positive term and the partial sums of odd index end with a negative term. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Absolute and Conditional Convergence Estimating the Sum of an Alternating Series You have seen that if a0, a1, a2, . . . is a decreasing sequence of positive numbers that tends to 0, then converges to some sum L. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Taylor Polynomials in x; Taylor Series in x Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Taylor Polynomials in x; Taylor Series in x The following estimate for Rn(x) is an immediate consequence of Corollary 12.6.2: where J is the closed interval that joins 0 to x. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Taylor Polynomials in x; Taylor Series in x By definition 0! = 1. Adopting the convention that f (0) = f , we can write Taylor polynomials in Σ notation: In this case, we say that f (x) can be expanded as a Taylor series in x and write Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Taylor Polynomials in x; Taylor Series in x Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Taylor Polynomials and Taylor Series in x – a Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Taylor Polynomials and Taylor Series in x – a The polynomial is called the nth Taylor polynomial for g in powers of x − a. In this more general setting, the Lagrange formula for the remainder, Rn(x), takes the form where c is some number between a and x. Now let x  I, x ≠ a, and let J be the closed interval that joins a to x. Then Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Taylor Polynomials and Taylor Series in x – a If Rn(x) → 0, then we have the series representation which, in sigma notation, takes the form This is known as the Taylor expansion of g(x) in powers of x − a. The series on the right is called a Taylor series in x − a. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Power Series Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Power Series There are exactly three possibilities for a power series: Case 1. The series converges only at x = 0. This is what happens with For x ≠ 0, kk xk 0, and so the series cannot converge. Case 2. The series converges absolutely at all real numbers x. This is what happens with the exponential series Case 3. There exists a positive number r such that the series converges absolutely for |x| < r and diverges for |x| > r . This is what happens with the geometric series Here there is absolute convergence for |x| < 1 and divergence for |x| > 1. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Power Series Associated with each case is a radius of convergence: In Case 1, we say that the radius of convergence is 0. In Case 2, we say that the radius of convergence is ∞. In Case 3, we say that the radius of convergence is r. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentiation & Integration of Power Series Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentiation & Integration of Power Series Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentiation & Integration of Power Series Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentiation & Integration of Power Series Term-by-term integration can be expressed by writing Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentiation & Integration of Power Series Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentiation & Integration of Power Series Power Series; Taylor Series It is time to relate Taylor series to power series in general. The relationship is very simple. Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.