Sequences and Series¶

Sequence¶

A sequence is an ordered list of numbers (e.g., $a_n$ ); the numbers are called “elements” or “terms”. Every convergent sequence is bounded, thus an unbounded sequence is divergent.

Sequence Test	Converge	Notes
Squeeze Theorem	$\lim\limits_{n \to \infty} a_n = \lim\limits_{n \to \infty} c_n = L$ then $\lim\limits_{n \to \infty} b_n = L$	$a_n \le b_n \le c_n$
Def 1, pg 692	$\lim\limits_{n \to \infty} a_n = L$
l’Hospital’s Rule	$\lim\limits_{n \to \infty} \frac{f(x)}{g(x)} \Rightarrow \lim\limits_{n \to \infty} \frac{f'(x)}{g'(x)}$	where $f(x)$ = numerator and $g(x)$ = denominator
Theorem 3, pg 693	if $f(n) = a_n$ then $\lim\limits_{n \to \infty} f(x)=L$
Theorem 6, pg 694	$\lim\limits_{n \to \infty} \| a_n \| =0$ then $a_n$ converges
Theorem 9, pg 696	$\lim\limits_{n \to \infty} r^n = \begin{cases} 0, & \text{if } -1 < r < 1 \\ 1, & \text{if } r = 1 \end{cases}$	Divergent for all other values of $r$
Theorem 12, pg 698	Every bounded ( $m \le a_n \le M$ ), monotonic sequence is convergent	The bounds exists for $n \ge 1$ , also see Theorem 10 and 11

Series¶

A series is the sum of the terms of a sequence: $\sum\limits_{n=1}^\infty a_n$ .

Series Test	Converge	Diverge	Notes
Divergence	N/A	$\lim\limits_{n\to\infty} a_n \ne 0$	Doesn’t show convergence and the converse is not true
Integral	if $\int\limits_1^\infty f(x) dx$ converges	if $\int\limits_1^\infty f(x) dx$ diverges	$f(x)$ must be positive, decreasing, and continous, also $f(n) = a_n \text{ for all } n$
Root	$\lim\limits_{n\to\infty}\sqrt[n]{\|a_n\|} = L < 1$	$\lim\limits_{n\to\infty}\sqrt[n]{\|a_n\|} = L > 1 \text{ or } \infty$	inconclusive if $L = 1$
Ratio	$\lim\limits_{n\to\infty} \left\| \frac{a_{n+1}}{a_n}\right\| = L < 1$	$\lim\limits_{n\to\infty} \left\| \frac{a_{n+1}}{a_n}\right\| = L > 1 \text{ or } \infty$	inconclusive if $L = 1$
Direct Comparison	$0 \le a_n \le b_n \text{ for all } n$ and $\sum\limits_{n=1}^{\infty} b_n$ converges	$0 \le b_n \le a_n \text{ for all } n$ and $\sum\limits_{n=1}^{\infty} b_n$ diverges	$a_n,b_n > 0$
Limit Comparison	$\lim\limits_{n\to\infty} \frac{a_n}{b_n} = L$ and $\sum\limits_{n=1}^{\infty} b_n$ converges	$\lim\limits_{n\to\infty} \frac{a_n}{b_n} = L$ and $\sum\limits_{n=1}^{\infty} b_n$ diverges	$a_n,b_n > 0$ and L is a positive constant, if L is $\infty$ or 0, then pick a different $b_n$
Absolute	$\sum\limits_{n=1}^{\infty} \| a_n \| = 0$		Definition of absolutely convergent, the sum is independent of the order in which the terms are summed
Conditional	$\sum\limits_{n=1}^{\infty} \| a_n \|$ diverges but $\sum\limits_{n=1}^{\infty} a_n$ converges		The sum is dependent of the order in which the terms are summed

Common Series¶

Series Test	Formula	Converge	Diverge	Notes
Alternating	$\sum\limits_{n=1}^\infty (-1)^{n-1} a_n$	$0 < a_{n+1} \le a_n \text{ for all } n$ and $\lim\limits_{n \to \infty} a_n = 0$	N/A
Geometric	$\sum\limits_{n=1}^\infty ar^{n-1}$	$\|r\| < 1$ and converges to $\frac{a}{1-r}$	$\|r\| \ge 1$	finite sum of the first n terms: $= \frac{a(1-r^n)}{1-r}$
P-Series	$\sum\limits_{n=1}^\infty \frac{1}{n^p}$	$p > 1$	$p \le 1$	cannot calculate sum
Power	$\sum\limits_{n=0}^\infty c_n (x-a)^n$	$\begin{array}{l} i, \text{converge if } x=a \\ ii, \text{converge for all } x \\ iii, \text{converge if } \|x-a\|<R \end{array}$		$R$ is the radius of convergence, you need to check the end points for convergence too. Typically use Ratio Test.
Taylor	$\sum\limits_{n=0}^\infty \frac{f^n (a)}{n!} (x-a)^n$	$\|x-a\|<R$	$\|x-a\|>R$	Taylor series is centered about a. Same note as power series
Maclaurin	$\sum\limits_{n=0}^\infty \frac{f^n (0)}{n!} (x)^n$	$\|x\|<R$	$\|x\|>R$	A Macluarin series is a Taylor series centered about 0. Same note as power series