Series

A series is the sum of the elements in a (in)finite sequence

\sum_{n = 1}^{\infty} a_{n} = a_{1} + a_{2} + \dots

Partial Sums

The $N^{t h}$ partial sum of the series $\sum a_{n}$ is the sum of the first $N$ terms of other series

S_{N} = \sum_{n = 1}^{N} a_{n} = a_{1} + a_{2} + \dots + a_{N}

We can use sequences of partial sums to analyze the original series of a sequence

S_{1}, S_{2}, S_{3} \dots = {S_{N}}_{N = 1}^{\infty}

lim_{n \to \infty} S_{N} = S \to \sum_{n = 1}^{\infty} a_{n} converges to S

If the sequence of partial sums doesn't have a finite limit, the series diverges

Convergence of Series

Geometric Series

\sum_{n = 0}^{\infty} a r^{n} = a + a r + a r^{2} + \dots

$| r | < 1 \to \sum a_{n}$ converges to $\frac{a}{1 - r}$
$| r | \geq 1 \to \sum a_{n}$ diverges
(when the series starts at $n = 0$ )

Using infinite sum rules to solve the limits of series
$\sum_{n = 1}^{\infty} \frac{2^{2 n - 1}}{3^{n}} = \sum_{n = 0}^{\infty} \frac{1}{3} {(\frac{2}{3})}^{n}$
$\frac{a}{1 - r} = 1$

Geometric series up to $x :$

\sum_{n = 0}^{x} a r^{n} = \frac{a (1 - r^{x + 1})}{1 - r}

P-Series

\sum_{n = 1}^{\infty} \frac{1}{n^{p}}, p > 0

Series converges for $p > 1$ and diverges for $0 < p \leq 1$

Collapsing/Telescoping Series

A series in which most terms cancel out

\sum_{n = 2}^{\infty} \frac{1}{n - 1} - \frac{1}{n} = 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} + \dots + \frac{1}{n} - \frac{1}{n + 1}

lim_{n \to \infty} S_{n} = 1 - \frac{1}{n + 1} = 1