==Infinite series==
The idea that an infinite number of additions lead to a finite number was intial seen as unsettling (see [[Zeno's paradox]]) however it is straigtforward to see why this could happen,
Let <math>S=\sum^{\infty}_{n=1}\frac{1}{2^{n}}</math>
<math>S=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots</math>
<math>2S=1+\frac{1}{2}+\frac{1}{4}+\dots</math>
<math>2S=1+S</math>
<math>S=1</math>
This leads to an important class of infinties series called the geometric series.
===Geometric series===
<math>\sum^{\infty}_{n=1}r^{n}=\frac{1}{1-r},\qquad \mathrm{for}\ |r|<1.</math>
Where <math>|r|</math> denotes the [[absolute value]] of ''r''.
===Convergence===
However all such series don't converge,
<math>\sum^{\infty}_{n}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\dots</math>
<math>\leq 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\dots</math>
<math>=1++\frac{1}{2}++\frac{1}{2}+\dots</math>
Adding together an infinte string of halves will not give you a finite number so this sequence is said to '''diverge'''.
==Usage==
[[Category:mathemtics]]