List of series

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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[change | change source]

Where \zeta(s)\, is the Riemann zeta function.

Power series[change | change source]

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[change | change source]

  • \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[change | change source]

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[change | change source]

  • \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[change | change source]

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:

Binomial coefficients[change | change source]

  • \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[change | change source]

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[change | change source]

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

Related pages[change | change source]

Notes[change | change source]

  1. 1.0 1.1 1.2 1.3 Theoretical computer science cheat sheet

References[change | change source]