Geometric Series:

The series $\sum_{n = 0}^{\infty} a_{n}$ is said to be a geometric series if each term after the first is a fixed multiple of the term immediately before it. That is, there is a number $r$ , called the ratio of the series, such that

$a_{n + 1} = r a_{n}$ , for all $n \geq 0$

i.e. $a_{n} = a_{o} r^{n}$ and $\sum_{n = 0}^{\infty} a_{n} = a_{0} + r a_{0} + r^{2} a_{0} + r^{3} a_{0} + \dots$

Theorem: If $| r | < 1$ , then the geometric series $\sum_{n = 0}^{\infty} a_{n}$ converges and its sum is $S = \sum_{n = 0}^{\infty} a_{0} r^{n} = \frac{a_{0}}{1 - r}$ .
If $| r | > 1$ and $a_{0} \neq 0$ , then the geometric series diverges.
For example the radius of converges of the geometric series $\sum_{n = 0}^{\infty} {(\frac{1}{2})}^{2}$ is given by $r = \frac{1}{2} < 1$ , thus the series converges to $r = \frac{1}{1 (\frac{1}{2})} = 2$ .

Definition: If the sequence of partial sums of the series $\sum_{n = 0}^{\infty} a_{n}$ diverges to infinity, then we say that the series diverges to infinity and we write $\sum_{n = 0}^{\infty} a_{n} = \infty$ .

Other Famous Series:

The harmonic series is the series
$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} \dots = \sum_{n = 1}^{\infty} \frac{1}{n} .$ The harmonic series is divergent.
An alternating series is a series where terms alternate signs.
Example:
$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dots = \sum_{n = 1}^{\infty} {(- 1)}^{n + 1} \frac{1}{n} .$ This converges since $a_{n + 1} \leq a_{n}$ for all $n$ and $lim_{n \to \infty} a_{n} = 0$ .
The series $\sum_{n = 1}^{\infty} p^{n}$ is called the p-series. It converges $if p > 1$ and diverges if $p \leq 1$ .
For a given power series $\sum_{n = 0}^{\infty} c_{n} {(x - a)}^{n}$ there are only three possibilities:
1. The series converges only when $x = a$ .
2. The series converges for all x.
3. There is a positive number R such that the series converges if $| x - a | < R$ and diverges if $| x - a | > R$ .
  The number R in case (c) is called a radius of convergence of the power series. The interval of convergence of a power series is defined to be the interval consisting of all values of x for which the series converges.