Changes

==Infinite series==

==Infinite series==

''For more detail see [[Infinite series]].''

''For more detail see [[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,

 

The idea that an infinite number of additions lead to a finite number was intial seen as problematic or even impossible, (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>

:Let <math>S=\sum^{\infty}_{n=1}\frac{1}{2^{n}}</math>

:<math>S=1</math>

:<math>S=1</math>

This leads to an important class of infinties series called the geometric series.

This one of an important class of infinties series called the geometric series.

===Geometric series===

===Geometric series===

===Convergence===

===Convergence===

However all infinite series don't converge,

However all infinite series don't converge, a famous example is,

:<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>\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>