Geometric Series

In mathematics, a geometric series is a series with a constant ratio that is common between successive terms.

Definition

Any series which can be written in the following forms is a geometric series. The series is generated by multiplying each term by a common ratio.

$$ \sum_{n=1}^{\infty} ar^{n-1} \quad \sum_{n=0}^{\infty} ar^n $$

in which $a$ and $r$ are fixed real numbers, and $a \ne 0$

__Note:__ These two series are identical, except for a _shift in the index variable_ from $1$ to $0$
(not unlike the difference between a 0-indexed array and a 1-indexed array in computer science):

Common Ratio

The common ratio is found by dividing any two consecutive terms: $\frac{a_n+1}{a_n}$

Examples

$$ \sum_{n=0}^{\infty} {a_n} = \left\{ (1)(2)^n \right\} = 1 + 2 + 3 + 4 + ... + a_n $$$$ \sum_{n=1}^{\infty} {b_n} = \left\{ (1)\left( \frac{1}{2} \right)^{n-1} \right\} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... + b_n $$$$ \sum_{n=1}^{\infty} {c_n} = \left\{ (-1)^{n+1}\left(\frac{1}{2}\right)^{n} \right\} = 1 -\frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} - \frac{1}{32} + ... - c_n $$

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Convergence & Divergence of Geometric Series

Convergent

A series is convergent when the following conditions apply:

If $|r| < 1$ then $r^n \to 0$ as $n \to \infty$ and the sequence of partial sums $s_n \to \frac{a}{1-r} \\$.

$$\sum_{n=1}^{\infty} a\cdot r^{n-1} = \frac{a}{1 - r} \quad \text{ IF } |r| < 1$$

Divergent

A series is divergent when these conditions apply:

When $|r| \ge 1$ then $|r^n| \to \infty$, and the sequence of partial sums has no finite limit.