Editing Series (mathematics) (section)

==Examples of numerical series==
{{For|other examples|List of mathematical series|Sums of reciprocals#Infinitely many terms}}
* A ''[[geometric series]]''<ref name=":45" /><ref name=":24" /> is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). For example: <math display=block>
1 + {1 \over 2} + {1 \over 4} + {1 \over 8} + {1 \over 16} + \cdots=\sum_{n=0}^\infty{1 \over 2^n} = 2.
</math> In general, a geometric series with initial term <math> a</math> and common ratio <math> r</math>, <math display=inline>\sum_{n=0}^\infty a r^n,</math> converges if and only if <math display=inline>|r| < 1</math>, in which case it converges to <math display=inline> {a \over 1 - r}</math>.
* The ''[[harmonic series (mathematics)|harmonic series]]'' is the series<ref name=":243">{{harvnb|Apostol|1967|p=384}}</ref> <math display=block>1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots = \sum_{n=1}^\infty {1 \over n}.</math> The harmonic series is [[harmonic series (mathematics)#Divergence|divergent]].
* An ''[[alternating series]]'' is a series where terms alternate signs.<ref name=":2434">{{harvnb|Apostol|1967|pp=403–404}}</ref> Examples: <math display=block>
1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots
= \sum_{n=1}^\infty {\left(-1\right)^{n-1} \over n} = \ln(2),
</math> the [[alternating harmonic series]], and <math display=block>
-1+\frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + \cdots
= \sum_{n=1}^\infty \frac{\left(-1\right)^n}{2n-1} = -\frac{\pi}{4},
</math> the [[Leibniz formula for π|Leibniz formula for <math>\pi.</math>]]
* A [[telescoping series]]<ref name=":2432">{{harvnb|Apostol|1967|p=386}}</ref> <math display=block>
\sum_{n=1}^\infty \left(b_n-b_{n+1}\right)
</math> converges if the [[sequence]] {{tmath|b_n}} converges to a limit {{tmath|L}} as {{tmath|n}} goes to infinity. The value of the series is then {{tmath|b_1 - L}}.<ref name=":10">{{harvnb|Apostol|1967|p=387}}</ref>
* An ''[[arithmetico-geometric series]]'' is a series that has terms which are each the product of an element of an [[arithmetic progression]] with the corresponding element of a [[geometric progression]]. Example: <math display=block>3 + {5 \over 2} + {7 \over 4} + {9 \over 8} + {11 \over 16} + \cdots=\sum_{n=0}^\infty{(3+2n) \over 2^n}.</math>
* The [[Dirichlet series]] <math display=block>
\sum_{n=1}^\infty\frac{1}{n^p}
</math> converges for {{tmath|p>1}} and diverges for {{tmath|p \leq 1}}, which can be shown with the [[integral test for convergence]] described below in [[Series (mathematics)#Convergence tests|convergence tests]]. As a function of {{tmath|p}}, the sum of this series is [[Riemann zeta function|Riemann's zeta function]].<ref name=":2433">{{harvnb|Apostol|1967|p=396}}</ref>
* [[Hypergeometric series]]: <math display="block">
_pF_q \left[ \begin{matrix}a_1, a_2, \dotsc, a_p \\ b_1, b_2, \dotsc, b_q \end{matrix}; z \right]
:= \sum_{n=0}^{\infty} \frac{\prod_{r=1}^{p} (a_r)_n}{\prod_{s=1}^{q} (b_s)_n} \frac{z^n}{n!}
</math> and their generalizations (such as [[basic hypergeometric series]] and [[elliptic hypergeometric series]]) frequently appear in [[integrable systems]] and [[mathematical physics]].<ref>Gasper, G., Rahman, M. (2004). Basic hypergeometric series. [[Cambridge University Press]].</ref>
* There are some elementary series whose convergence is not yet known/proven. For example, it is unknown whether the Flint Hills series, <math display=block>
\sum_{n=1}^\infty \frac{1}{n^{3}\sin^{2} n},
</math> converges or not. The convergence depends on how well <math>\pi</math> can be approximated with [[rational numbers]] (which is unknown as of yet). More specifically, the values of {{tmath|n}} with large numerical contributions to the sum are the numerators of the continued fraction convergents of <math>\pi</math>, a sequence beginning with 1, 3, 22, 333, 355, 103993, ... {{OEIS|A046947}}. These are integers {{tmath|n}} that are close to <math>m\pi</math> for some integer {{tmath|m}}, so that <math>\sin n</math> is close to <math>\sin m\pi = 0</math> and its reciprocal is large.

===Pi===
{{Main|Basel problem|Leibniz formula for π}}

<math display=block> \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}</math>

<math display=block> \sum_{n=1}^\infty \frac{(-1)^{n+1}(4)}{2n-1} = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} - \cdots = \pi</math>

===Natural logarithm of 2===
{{Main|Natural logarithm of 2#Series representations}}

<math display=block>\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \ln 2</math>

<math display=block> \sum_{n=1}^\infty \frac{1}{2^{n}n} = \ln 2</math>

===Natural logarithm base {{mvar|e}} ===
{{Main|e (mathematical constant)}}

<math display=block>\sum_{n = 0}^\infty \frac{(-1)^n}{n!} =  1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots = \frac{1}{e}</math>

<math display=block> \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots = e </math>