# Calculus/Definition of a Series

## Definition of a Series

Series
A series is the sum of the terms in a sequence.

For a sequence ${\displaystyle d}$, the series ${\displaystyle D}$ would be ${\displaystyle D=d_{1}+d_{2}+d_{3}...}$. This is true for all series, as it follows from the definition. Only adding a sub-sequence is called a partial sum.

## Summation Notation

Purely using the prior definition of a series is possible, but unwieldy. Instead we can again put to use summation notation, which was partially covered in the section on 'integrals'. Some common properties and identities are outlined here.

### Identities

${\displaystyle \sum _{k=0}^{n}{c}=nc}$ where ${\displaystyle c}$ is some constant.
${\displaystyle \sum _{k=0}^{n}{k}={\frac {n(n+1)}{2}}}$
${\displaystyle \sum _{k=0}^{n}{k^{2}}={\frac {n(n+1)(2n+1)}{6}}}$
${\displaystyle \sum _{k=0}^{n}{k^{3}}={\frac {n^{2}(n+1)^{2}}{4}}}$

### Properties

${\displaystyle \sum _{k}^{n}{s_{k}}+\sum _{n}^{m}{s_{k}}=\sum _{k}^{m}{s_{k}}}$ This is the adding of sums.
${\displaystyle j\sum _{k}^{n}{s_{k}}=\sum _{k}^{n}{js_{k}}}$ Note that this is essentially the distributive property, so this will work for anything that follows the distributive property, even non-constant terms.