kids encyclopedia robot

List of series facts for kids

Kids Encyclopedia Facts


This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums.

Sums of powers

  • \sum_{i=1}^n i = \frac{n(n+1)}{2}\,\!
    See also triangle number. This is one of the most useful series: many applications can be found throughout mathematics.
  • \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6} \,\!
  • \sum_{i=1}^n i^3 = \left[\frac{n(n+1)}{2}\right]^2 = \frac{n^4}{4} + \frac{n^3}{2} + \frac{n^2}{4} = \left(\sum_{i=1}^n i\right)^2\,\!
  • \sum_{i=1}^{n} i^{4} = \frac{n(n+1)(2n+1)(3n^{2}+3n-1)}{30}=\frac{6 n^5+15 n^4+10 n^3-n}{30}\,\!
  • \sum_{i=0}^n i^s = \frac{(n+1)^{s+1}}{s+1} + \sum_{k=1}^s\frac{B_k}{s-k+1}{s\choose k}(n+1)^{s-k+1}\,\!
    Where B_k\, is the k\,th Bernoulli number, B_1\, is negative and s\choose k is the binomial coefficient (choose function).
  • \sum_{i=1}^\infty i^{-s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = \zeta(s)\,\!
Where \zeta(s)\, is the Riemann zeta function.

Power series

Infinite sum (for |x| < 1) Finite sum
\sum_{i=0}^\infty x^i= \frac{1}{1-x}\,\! \sum_{i=0}^n x^i = \frac{1-x^{n+1}}{1-x} = 1+\frac{1}{r}\left(1-\frac{1}{(1+r)^n}\right) where r>0 and x=\frac{1}{1+r}.\,\!
\sum_{i=0}^\infty x^{2i}= \frac{1}{1-x^2}\,\!
\sum_{i=1}^\infty i x^i = \frac{x}{(1-x)^2}\,\! \sum_{i=1}^n i x^i = x\frac{1-x^n}{(1-x)^2} - \frac{n x^{n+1}}{1-x}\,\!
\sum_{i=1}^{\infty} i^2 x^i =\frac{x(1+x)}{(1-x)^3}\,\! \sum_{i=1}^n i^2 x^i = \frac{x(1+x-(n+1)^2x^n+(2n^2+2n-1)x^{n+1}-n^2x^{n+2})}{(1-x)^3} \,\!
\sum_{i=1}^{\infty} i^3 x^i =\frac{x(1+4x+x^2)}{(1-x)^4}\,\!
\sum_{i=1}^{\infty} i^4 x^i =\frac{x(1+x)(1+10x+x^2)}{(1-x)^5}\,\!
\sum_{i=1}^{\infty} i^k x^i = \operatorname{Li}_{-k}(x),\,\! where Lis(x) is the polylogarithm of x.

Simple denominators

  • \sum^{\infty}_{n=1} \frac{x^n}n = \log_e\left(\frac{1}{1-x}\right) \quad\mbox{ for } |x| < 1 \!
  • \sum^{\infty}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots = \arctan(x)\,\!
  • \sum^{\infty}_{n=0} \frac{x^{2n+1}}{2n+1} = \mathrm{arctanh} (x) \quad\mbox{ for } |x| < 1\,\!
  • \sum^{\infty}_{n=1} \frac{1}{n^2} = \frac{\pi^2}{6}\,\!
  • \sum^{\infty}_{n=1} \frac{1}{n^4} = \frac{\pi^4}{90}\,\!
  • \sum^{\infty}_{n=1} \frac{y}{n^2+y^2} = -\frac{1}{2y}+\frac{\pi}{2}\coth(\pi y)

Factorial denominators

Many power series which arise from Taylor's theorem have a coefficient containing a factorial.

  • \sum^{\infty}_{i=0} \frac{x^i}{i!} = e^x
  • \sum^{\infty}_{i=0} \frac{(-1)^i}{(2i+1)!} x^{2i+1}= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sin x
  • \sum^{\infty}_{i=0} \frac{(-1)^i}{(2i)!} x^{2i} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \cos x
  • \sum^{\infty}_{i=0} \frac{x^{2i+1}}{(2i+1)!} = \sinh x
  • \sum^{\infty}_{i=0} \frac{x^{2i}}{(2i)!} = \cosh x

Modified-factorial denominators

  • \sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} = \arcsin x\quad\mbox{ for } |x| < 1\!
  • \sum^{\infty}_{i=0} \frac{(-1)^i (2i)!}{4^i (i!)^2 (2i+1)} x^{2i+1} = \mathrm{arcsinh}(x) \quad\mbox{ for } |x| < 1\!

Binomial series

Geometric series:

  • (1+x)^{-1} = \begin{cases} \displaystyle \sum_{i=0}^\infty (-x)^i & |x|<1 \\ \displaystyle \sum_{i=1}^\infty -(x)^{-i} & |x|>1 \\ \end{cases}

Binomial Theorem:

  • (a+x)^n = \begin{cases} \displaystyle \sum_{i=0}^\infty \binom{n}{i} a^{n-i} x^i & |x| \! < \! |a| \\ \displaystyle \sum_{i=0}^\infty \binom{n}{i} a^i x^{n-i} & |x| \! > \! |a| \\ \end{cases}
  • (1+x)^\alpha = \sum_{i=0}^\infty {\alpha \choose i} x^i\quad\mbox{ for all } |x| < 1 \mbox{ and all complex } \alpha\!
with generalized binomial coefficients
{\alpha\choose n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}\!

Square root:

  • \sqrt{1+x} = \sum_{i=0}^\infty \frac{(-1)^i(2i)!}{(1-2i)i!^24^i}x^i \quad\mbox{ for } |x|<1\!

Miscellaneous:

  • \sum_{i=0}^\infty {i+n \choose i} x^i = \frac{1}{(1-x)^{n+1}}
  • \sum_{i=0}^\infty \frac{1}{i+1}{2i \choose i} x^i = \frac{1}{2x}(1-\sqrt{1-4x})
  • \sum_{i=0}^\infty {2i \choose i} x^i = \frac{1}{\sqrt{1-4x}}
  • \sum_{i=0}^\infty {2i + n \choose i} x^i = \frac{1}{\sqrt{1-4x}}\left(\frac{1-\sqrt{1-4x}}{2x}\right)^n

Binomial coefficients

  • \sum_{i=0}^n {n \choose i} = 2^n
  • \sum_{i=0}^n {n \choose i}a^{(n-i)} b^i = (a + b)^n
  • \sum_{i=0}^n (-1)^i{n \choose i} = 0
  • \sum_{i=0}^n {i \choose k} = { n+1 \choose k+1 }
  • \sum_{i=0}^n {k+i \choose i} = { k + n + 1 \choose n }
  • \sum_{i=0}^r {r \choose i}{s \choose n-i} = {r + s \choose n}

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

  • \sum_{i=1}^n \sin\left(\frac{i\pi}{n}\right) = 0
  • \sum_{i=1}^n \cos\left(\frac{i\pi}{n}\right) = 0

Unclassified

  • \sum_{n=b+1}^{\infty} \frac{b}{n^2 - b^2} = \sum_{n=1}^{2b} \frac{1}{2n}

Related pages

  • Many books with a list of integrals also have a list of series.
kids search engine
All content from Kiddle encyclopedia articles (including the article images and facts) can be freely used under Attribution-ShareAlike license, unless stated otherwise. Cite this article:
List of series Facts for Kids. Kiddle Encyclopedia.