Series (mathematics)

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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.

Contents

Summation notation

If ever value in a sequence is denoted an, then the sum of N of these are denoted,

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

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

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

\sum_{n=500}^{1000}n

is the sum of the integers from 500 to 1000.

Finite series

For more detail see Finite series.

The sum of finite series is in general relatively straight forward to calculate it consist of merely summing together all the numbers in a sequences. However this can be time 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 allegorical leads to a key insight in finite sums,

\sum^{N}_{n=1}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 initial seen as problematic or even impossible, (see Zeno's paradox). However it is straightforward to see why this could happen,

Let S=\sum^{\infty}_{n=1}\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 one of an important class of infinites series called the geometric series.

Geometric series

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

Where | r | denotes the absolute value of r.

Convergence

For more detail see Convergence

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

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

This will converge if \lim_{n\rightarrow\infty}|\frac{a_{n+1}}{a_{n}}|<1.

Alternating Series convergence

An alternating series is of the form,

\sum^{\infty}_{n=1}(-1)^{n}a_{n}

This will converge if 0\leq a_{n+1}<a_{n}.

Usage

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