==Summation notation==
If ever value in a sequence is denoted <math>a_{n}</math>, then the sum of ''N'' of these are denoted,
<math>a_{1}+a_{2}+a_{3}+\dots+a_{N}=\sum^{N}_{n=1}a_{n}</math>
This is usal usually pronounced "the sum of 1 to N of a n" (note that this is still an ambigous ambiguous statement which is why mathematics is more riquerous rigorous written down).
If we are conserned concerned with particular number we can move the starting index. For example,
<math>\sum_{n=500}^{1000}n</math>
is the sum of the intergers integers from 500 to 1000.
==Finite series==
''For more detail see [[Finite series]].''
The sum of finite series is ingenral relativly in general relatively straight forward to calculate it consist of mearly merely summing together all the numbers in a sequences. However this can be time consumming consuming and in pre-computer mathematics many shortcuts were found.
Gauss once famously was given the assignment by his teacher to add the number between 1 and 100. The rest of the students were working hard adding carefully, but after less than a minute he correctly wrote 5050. This story whilst is allogorical allegorical leads to a key insight in finite sums,
:<math>\sum^{N}_{n=1}n=\frac{N(N+1)}{2}</math>
''For more detail see [[Infinite series]].''
The idea that an infinite number of additions lead to a finite number was intial initial seen as problematic or even impossible, (see [[Zeno's paradox]]). However it is straigtforward straightforward to see why this could happen,
:Let <math>S=\sum^{\infty}_{n=1}\frac{1}{2^{n}}</math>
:<math>S=1</math>
This one of an important class of infinties infinites series called the geometric series.
===Geometric series===
===Convergence===
''For more detail see [[Convergence]]''
However all infinite series don't converge, a famous example is,
:<math>=1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\dots</math>
Adding together an infinte infinite string of halves will not give you a finite number so this sequence is said to '''diverge'''.
A series will only converge if <math>\lim_{n\rightarrow\infty}a_{n}=0</math>. However , there are two other conditions for real series,
====Power Series convergence====
*[[Fourier series]]
[[Category:mathematicsMathematics]]