# Series

A series is the indicated sum of the terms of a sequence.

Example 1:

Finite sequence            : $3,7,11,15,19$

Related finite series      : $3+7+11+15+19$

Infinite sequence          : $\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\cdots$

Related infinite series   : $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots$

A series can be written in an abbreviated form by using the Greek letter $\sum$(sigma), called the summation sign.  For instance, to abbreviate the writing of the series $2+4+6+8+\cdots +50$ first notice that the general term of the series is $2n$.  The series begins with the term for $n=1$ and ends with the term for $n=25$.  Using sigma notation, you can write this series as $\sum _{n=1}^{25}2n$, which is read “the sum of $2n$ for values of $n$ from $1$ to $25$.”

$\begin{array}{l}\sum _{n=1}^{25}2n=2\cdot 1+2\cdot 2+2\cdot 3+2\cdot 4+\cdots +2\cdot 25\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=2+4+6+8+\cdots +50\end{array}$

There are formulas to help you quickly find the value of finite arithmetic series, finite geometric series, and certain kinds of infinite geometric series.