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 +
ar ^{2} + ar^{3} + ...
` (Example 1) converges to

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
`{a _{n}}` and and the sum of

Be able to apply the integral test to the "tail" (the sum from the

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
`c _{0} + c_{1}x + c_{2}x^{2} + ...`

8.6 A power series in `x` defines a function `f(x)`.
[Example 1: `1/(1+x ^{2}) = 1 - x^{2} + x^{4} - ...
`; 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

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)x ^{2}/2 + ... +
f^{(n)}(0)x^{n}/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 `e ^{x}, 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

8.8 The Binomial Series. Understand that the Binomial Theorem

`(a+b) ^{k} = a^{k} + C(k,1)a^{k-1}b
+ C(k,2)a^{k-2}b^{2} + ... +
C(k,k-2)a^{2}b^{k-2} + C(k,k-1)ab^{k-1}
+ b^{k}`

where