### MAT 132 Spring 2006 Review - Chapter 8

References are to Stewart, Single Variable Calculus - SBU Edition

8.1 Know what a sequence is, and be able to check convergence/divergence in simple cases: for rational functions of n as in Example 4 [Exercises 10, 11]; using Theorem 2 p.559 and l'Hôpital's rule [Example 5, Exercises 19, 21].

8.2 Understand that the sum of an infinite series only makes sense as the limit of the sequence of partial sums (Definition 2 p.568). Fundamental example: the geometric series a + ar + ar2 + ar3 + ... (Example 1) converges to a/(1-r) if |r| < 1 and diverges otherwise. Be able to sum this series correctly [Example 2, Exercises 11, 13, 15]. Also, be able to use this technique to express a repeating decimal as a fraction [Example 4, Exercises 31-33]. Second fundamental example: the harmonic series 1 + 1/2 + 1/3 + ..., which diverges (Example 7). This is fundamental because it shows that the terms can go to zero and the series may still diverge (Note 2 p. 572).

8.3 Be able to apply the integral test, and determine convergence or divergence of a series from covergence or divergence of the corresponding improper integral (discussion on pp. 577, 578) [Exercise 7].
Be able to apply the comparison test: suppose 2 series, the sum of {an} and and the sum of {bn} are related by the inequality 0 < an < bn for every n; then if the smaller series diverges the larger one must diverge, and if the larger one converges the smaller one must converge. [Examples 3, 4, Exercises 9, 17, 25].
Be able to apply the integral test to the "tail" (the sum from the (n+1)-st term on) of a series to estimate the error involved in approximating the infinite sum by the sum of the first n terms [Example 6, Example 7, Exercises 29, 31].

8.4 Understand that an alternating series (terms are alternately positive and negative) will converge if the terms are decreasing in absolute value and if their limit is 0 (Box, page 588) [Example 1, Exercises 3, 5, 7] and the "Alternating Series Estimation Theorem" (p. 588) [Example 4, Exercises 13, 15, 17].
Understand what absolute convergence means, and that it implies convergence [Example 7, Exercises 19, 27].
VERY IMPORTANT FOR POWER SERIES: Be able to apply the ratio test to a series (Box, p. 591) [Examples 8,9, Exercises 31, 33].

8.5 Power series. Understand what a power series is ("an infinite polynomial") and that in general the convergence of c0 + c1x + c2x2 + ... will depend on x. The Ratio test is very useful here [Examples 1, 2 ]. Understand the difference between the radius of convergence and the interval of convergence (p. 596) [Examples 4, 5, Exercises 5, 7],

8.6 A power series in x defines a function f(x). [Example 1: 1/(1+x2) = 1 - x2 + x4 - ... ; note that the domain of definition of the function (here it is the whole line) may be different from the interval of convergence of the series (here it is (-1, 1)! Exercises 5, 7, 9].
Be able to calculate new power series by term-by-term differentiation or integration of old ones (Theorem 2 p. 601) [Examples 5, 6, 7, Exercises 11, 13].

8.7 Taylor series. If a function f(x) has derivatives of every order at 0, then the power series

f(0) + f'(0)x + f''(0)x2/2 + ... + f(n)(0)xn/n! + ...

is called the Taylor series for f(x) about 0; also called the Maclaurin series for f(x). More generally if f(x) has derivatives of every order at x=a, then the power series

f(a) + f'(a)(x-a) + f''(a)(x-a)2/2 + ... + f(n)(a)(x-a)n/n! + ...

is called the Taylor series for f(x) about a [Examples 1, 3, Exercises 3, 5, 7]. Be able to apply Taylor's Inequality (p. 608) -you do not need to memorize it- to show that ex, sin x, cos x are equal to the sum of their Maclaurin series [Example 2, Exercise 15, 16]. Be able to use Taylor series to integrate functions that can't be handled by ordinary methods, like e-x2 [Example 8, Exercises 33, 35].

8.8 The Binomial Series. Understand that the Binomial Theorem
(a+b)k = ak + C(k,1)ak-1b + C(k,2)ak-2b2 + ... + C(k,k-2)a2bk-2 + C(k,k-1)abk-1 + bk
where C(k,n) is the binomial coefficient C(k,n) = [k(k-1)(k-2)...(k-n+1)]/n!, generalizes to cases where k is not a positive integer. This applies in particular to (1+x)k and gives the binomial series [Box, page 618]. When k is a positive integer this series terminates, as above. Otherwise it is a power series with radius of convergence equal to one. [Examples 1, 2, Exercises 1, 3, 7, 9].