Calculus/Definition of a Series

Definition of a Series

For a sequence $d$, the series $D$ would be $D=d_1+d_2+d_3...$. 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

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

Properties

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