A sequence is any function whose domain is $\{k,k+1,k+2,\dots\}$ for some integer k.

Definition of convergence of a sequence: A sequence {a_n} is said to converge to L if, for every $\epsilon > 0$, there exists an integer k such that $\vert a_n - L\vert < \epsilon$ for every $n \geq k$. In this case, we write $\lim_{n \rightarrow \infty} = L$.

Any sequence which does not converge is said to diverge.

Definition of divergence of a sequence to $\infty$: A sequence {a_n} is said to diverge to $\infty$ if for every M, there exists an integer k such that a_n > M for all $n \geq k$.

A series is a formal sum of infinitely many terms. If the terms are $a_0, a_1, a_2, \dots$, then the series is typically denoted $\sum_{n=0}^{\infty} a_n$.

Given a series $\sum_{n=0}^{\infty} a_n$, the n-th partial sum S_n is defined by $S_n = a_0 + a_1 + \cdots + a_n$.

Definition of convergence of a series: A series is said to converge if the sequence {S_n} converges.

A series $\sum a_n$ is said to converge absolutely if $\sum \vert a_n\vert$ converges.

If a and r are real numbers, the series $\sum_{n=0}^{\infty} ar^n$ is called a geometric series. $\sum_{n=0}^{\infty} ar^n$ converges if and only if -1 < r < 1. If -1 < r < 1, then $\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$ (you should be able to prove this).

If p is any real number, $\sum_{n=1}^{\infty} \frac{1}{n^p}$ is called a p-series with exponent p. A p-series converges if and only if p > 1. You should know how to prove this (using the integral test).

nth term test for divergence: If $\lim_{n \rightarrow \infty} a_n$ is not equal to zero, then $\sum a_n$ does not converge.

Comparison test: Let c_n be a sequence of nonnegative numbers.

• If $\sum c_n$ converges, and $0 \leq a_n \leq c_n$ for all (large enough) n, then $\sum a_n$ converges.
• If $\sum c_n$ diverges, and $a_n \geq c_n$ for all (large enough) n, then $\sum a_n$ diverges.

Limit Comparison test: Suppose you have two series $\sum a_n$ and $\sum b_n$, and $\lim_{n \to \infty} \frac{a_n}{b_n} = L$. If $0< L < \infty$, then

• if $\sum a_n$ converges, $\sum b_n$ converges, and
• if $\sum a_n$ diverges then $\sum b_n$ diverges.

Note: If you are using the limit comparison test on a series whose terms are the quotient of two polynomials, then use the limit comparison test with the series whose terms are the highest power of n in the numerator, divided by the highest power of n in the denominator.

Ratio test: Let a_n be a sequence of nonnegative numbers.

• if $\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}$ exists and is less than 1, then $\sum a_n$ converges
• if $\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}$ does not exist or if it exists and is greater than 1, then $\sum a_n$ diverges.

Note that the ratio test does not give any information if $\lim \frac{a_{n+1}}{a_n} = 1$. Also note that the ratio test does not give any information about what the series converges to, only whether or not it converges.

Alternating series:

A series $\sum a_n$ is said to be an alternating series if its terms alternate in sign.

Important fact about alternating series: If a_n is an (eventually) nonincreasing sequence of positive numbers, and if $\lim_{n} a_n = 0$, then the alternating series $\sum_{n=0}^{\infty} (-1)^n a_n$ converges.

Even more interesting than the result itself is the following byproduct of the proof: if E_n denotes the error associated with estimating the sum of the series from the nth partial sum, then E_n < |a_n+1|. In other words, the error incurred by using a partial sum is less than the first ommitted term.

Taylor polynomials: Let f be a function which is smooth (i.e. infinitely differentiable) at x=a. Define P_n, the Taylor polynomial degree n for f about x=a by

$$P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots + \frac{f{(n)}(a)}{n!}(x-a)^n.$$

Note that P_n has the following property: P_n^(k)(a) = f^(k)(a) for all $k \leq n$.

Taylor series: Let f be a function which is smooth (i.e. infinitely differentiable) at x=a. Define the Taylor series for f about x=a to be the series

$$f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$$

The interval of convergence of the Taylor series is the set of all values of x for which the Taylor series converges. Note that the interval of convergence always includes a. It turns out that the interval of convergence is an interval centerd at a. Thus it must take one of the following forms:

1.

(a-R,a+R] for some R;

a-R,a+R

for some R;

2.

[a-R,a+R) for some R;

3.

(a-R,a+R) for some R;

4.

$(-\infty,\infty)$ (i.e. all real numbers)

The number R is called the radius of convergence. Further, the convergence is absolute in the interval (a-R,a+R). In other words, if you have a Taylor series (about x=a) with radius of convergence R, the series converges absolutely for all $x \in (a-R,a+R)$, and it diverges for all x outside the interval [a-R,a+R]. At the endpoints, a+R and a-R, it may converge absolutely, it may converge (but not absolutely), or it may diverge.

The ratio test is a useful tool for computing the radius of convergence of a Taylor series.

Note that Taylor series can be added, subtracted, and differentiated termwise within their intervals of convergence (see section 5.1 of your book).

Applications of Taylor series:

• Estimating function values. For example, we could estimate $\sqrt{e}$ by noting that $\sqrt{e}=e^{1/2}=1 + (1/2) + \frac{(1/2)^2}{2!} + \frac{(1/2)^3}{3!} + \cdots$.
• Finding Taylor series for more complicated functions.