Series (mathematics)

In mathematics a series is the sum of a sequence of numbers. This article is intended to give the reader some understanding on the summation of finite series and why some infinite series converge whilst others diverge.

Summation notation

If ever value in a sequence is denoted $a_{n}$, then the sum of N of these are denoted,

$a_{1}+a_{2}+a_{3}+\dots +a_{N}=\sum _{n=1}^{N}a_{n}$

This is usal pronounced "the sum of 1 to N of a n" (note that this is still an ambigous statement which is why mathematics is more riquerous written down).

If we are conserned with particular number we can move the starting index. For example,

$\sum _{n=500}^{1000}n$

is the sum of the intergers from 500 to 1000.

Finite series

For more detail see Finite Series.

The sum of finite series is ingenral relativly straight forward to calculate it consist of mearly summing together all the numbers in a sequences. However this can be time consumming 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 leads to a key insight in finite sums,

$\sum _{n=1}^{N}n={\frac {N(N+1)}{2}}$

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,

Let $S=\sum _{n=1}^{\infty }{\frac {1}{2^{n}}}$

$S={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\dots$

$2S=1+{\frac {1}{2}}+{\frac {1}{4}}+\dots$

$2S=1+S$

$S=1$

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

Geometric series

$\sum _{n=1}^{\infty }r^{n}={\frac {1}{1-r}},\qquad \mathrm {for} \ |r|<1.$

Where $|r|$ denotes the absolute value of r.

Convergence

However all infinite series don't converge,

$\sum _{n}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+\dots$

$\geq 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+\dots$

$=1+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+\dots$

Adding together an infinte string of halves will not give you a finite number so this sequence is said to diverge.

A series will only converge if $\lim _{n\rightarrow \infty }a_{n}=0$. However there are two other conditions for real series,

Power Series convergence

A power series is of the form,

$\sum _{n=1}^{\infty }a_{n}=\sum _{n=1}^{\infty }b_{n}(x-a)^{n}$,

This will converge if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\rightarrow\infty}|\frac{a_{n+1}}{a_{n}}|<1} .

Alternating Series convergence

An alternating series is of the form,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum^{\infty}_{n=1}(-1)^{n}a_{n}}

This will converge if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq a_{n+1}<a_{n}} .

Usage

• Integral
• Riemann Zeta function
• Taylor series
• Fourier series