List of mathematical 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[edit]

See Faulhaber's formula.

  • \sum_{k=0}^m k^{n-1}=\frac{B_n(m+1)-B_n}{n}\,\!

The first few values are:

  • \sum_{k=1}^m k=\frac{m(m+1)}{2}\,\!
  • \sum_{k=1}^m k^2=\frac{m(m+1)(2m+1)}{6}=\frac{m^3}{3}+\frac{m^2}{2}+\frac{m}{6}\,\!
  • \sum_{k=1}^m k^3

=\left[\frac{m(m+1)}{2}\right]^2=\frac{m^4}{4}+\frac{m^3}{2}+\frac{m^2}{4}\,\!

See zeta constants.

  • \zeta(2n)=\sum^{\infty}_{k=1} \frac{1}{k^{2n}}=(-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}

The first few values are:

  • \zeta(4)=\sum^{\infty}_{k=1} \frac{1}{k^4}=\frac{\pi^4}{90}\,\!
  • \zeta(6)=\sum^{\infty}_{k=1} \frac{1}{k^6}=\frac{\pi^6}{945}\,\!

Power series[edit]

Low-order polylogarithms[edit]

Finite sums:

  • \sum_{k=1}^n k z^k = z\frac{1-(n+1)z^n+nz^{n+1}}{(1-z)^2}\,\!
  • \sum_{k=1}^n k^2 z^k = z\frac{1+z-(n+1)^2z^n+(2n^2+2n-1)z^{n+1}-n^2z^{n+2}}{(1-z)^3} \,\!
  • \sum_{k=1}^n k^m z^k = \left(z \frac{d}{dz}\right)^m \frac{z-z^{n+1}}{1-z}

Infinite sums, valid for |z|<1 (see polylogarithm):

  • \operatorname{Li}_n(z)=\sum_{k=1}^{\infty} \frac{z^k}{k^n}\,\!

The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form:

  • \frac{d}{dz}\operatorname{Li}_n(z)=\frac{\operatorname{Li}_{n-1}(z)}{z}\,\!
  • \operatorname{Li}_{1}(z)=\sum_{k=1}^\infty \frac{z^k}{k}=-\ln(1-z)\!
  • \operatorname{Li}_{0}(z)=\sum_{k=1}^\infty z^k=\frac{z}{1-z}\!
  • \operatorname{Li}_{-1}(z)=\sum_{k=1}^\infty k z^k=\frac{z}{(1-z)^2}\,\!
  • \operatorname{Li}_{-2}(z)=\sum_{k=1}^\infty k^2 z^k=\frac{z(1+z)}{(1-z)^3}\,\!
  • \operatorname{Li}_{-3}(z)=\sum_{k=1}^\infty k^3 z^k =\frac{z(1+4z+z^2)}{(1-z)^4}\,\!
  • \operatorname{Li}_{-4}(z)=\sum_{k=1}^\infty k^4 z^k =\frac{z(1+z)(1+10z+z^2)}{(1-z)^5}\,\!

Exponential function[edit]

  • \sum_{k=0}^\infty \frac{z^k}{k!} = e^z\,\!
  • \sum_{k=0}^\infty k^3 \frac{z^k}{k!} = (z + 3z^2 + z^3) e^z\,\!
  • \sum_{k=0}^\infty k^4 \frac{z^k}{k!} = (z + 7z^2 + 6z^3 + z^4) e^z\,\!
  • \sum_{k=0}^\infty k^n \frac{z^k}{k!} = z \frac{d}{dz} \sum_{k=0}^\infty k^{n-1} \frac{z^k}{k!}\,\! = e^z T_{n}(z)

where T_{n}(z) is the Touchard polynomials.

Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions[edit]

  • \sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)!}=\sin z\,\!
  • \sum_{k=0}^\infty \frac{z^{2k+1}}{(2k+1)!}=\sinh z\,\!
  • \sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{(2k)!}=\cos z\,\!
  • \sum_{k=0}^\infty \frac{z^{2k}}{(2k)!}=\cosh z\,\!
  • \sum_{k=1}^\infty \frac{(-1)^{k-1}(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\tan z, |z|<\frac{\pi}{2}\,\!
  • \sum_{k=1}^\infty \frac{(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\tanh z, |z|<\frac{\pi}{2}\,\!
  • \sum_{k=0}^\infty \frac{(-1)^k2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\cot z, |z|<\pi\,\!
  • \sum_{k=0}^\infty \frac{2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\coth z, |z|<\pi\,\!
  • \sum_{k=0}^\infty \frac{(-1)^{k-1}(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\csc z, |z|<\pi\,\!
  • \sum_{k=0}^\infty \frac{-(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\operatorname{csch} z, |z|<\pi\,\!
  • \sum_{k=0}^\infty \frac{(-1)^kE_{2k}z^{2k}}{(2k)!}=\sec z, |z|<\frac{\pi}{2}\,\!
  • \sum_{k=0}^\infty \frac{E_{2k}z^{2k}}{(2k)!}=\operatorname{sech} z, |z|<\frac{\pi}{2}\,\!
  • \sum_{k=0}^\infty \frac{(2k)!z^{2k+1}}{2^{2k}(k!)^2(2k+1)}=\arcsin z, |z|\le1\,\!
  • \sum_{k=0}^\infty \frac{(-1)^k(2k)!z^{2k+1}}{2^{2k}(k!)^2(2k+1)}=\operatorname{arsinh} {z}, |z|\le1\,\!
  • \sum_{k=0}^\infty \frac{(-1)^kz^{2k+1}}{2k+1}=\arctan z, |z|<1\,\!
  • \sum_{k=0}^\infty \frac{z^{2k+1}}{2k+1}=\operatorname{arctanh} z, |z|<1\,\!
  • \ln2+\sum_{k=1}^\infty \frac{(-1)^{k-1}(2k)!z^{2k}}{2^{2k+1}k(k!)^2}=\ln\left(1+\sqrt{1+z^2}\right), |z|\le1\,\!

Modified-factorial denominators[edit]

  • \sum^{\infty}_{n=0} \frac{\prod_{k=0}^{n-1}(4k^2+\alpha^2)}{(2n)!} z^{2n} + \sum^{\infty}_{n=0} \frac{\alpha \prod_{k=0}^{n-1}[(2k+1)^2+\alpha^2]}{(2n+1)!}  z^{2n+1} = e^{\alpha \arcsin{z}}, |z|\le1

Binomial coefficients[edit]

Harmonic numbers[edit]

  •  \sum_{k=1}^\infty H_k z^k = \frac{-\ln(1-z)}{1-z}, |z|<1
  •  \sum_{k=1}^\infty \frac{H_k}{k+1} z^{k+1} = \frac{1}{2}\left[\ln(1-z)\right]^2, \qquad |z|<1

Binomial coefficients[edit]

  • \sum_{k=0}^n {n \choose k} = 2^n
  • \sum_{k=0}^n (-1)^k {n \choose k} = 0
  • \sum_{k=0}^n {k \choose m} = { n+1 \choose m+1 }

Trigonometric functions[edit]

Sums of sines and cosines arise in Fourier series.

  • \sum_{k=1}^\infty \frac{\sin(k\theta)}{k}=\frac{\pi-\theta}{2}, 0<\theta<2\pi\,\!
  • \sum_{k=1}^\infty \frac{\cos(k\theta)}{k}=-\frac{1}{2}\ln(2-2\cos\theta), \theta\in\mathbb{R}\,\!
  • \sum_{k=0}^\infty \frac{\sin[(2k+1)\theta]}{2k+1}=\frac{\pi}{4}, 0<\theta<\pi\,\!
  • \sum_{k=0}^n \sin(\theta+k\alpha)=\frac{\sin\frac{(n+1)\alpha}{2}\sin(\theta+\frac{n\alpha}{2})}{\sin\frac{\alpha}{2}}\,\!
  • \sum_{k=1}^{n-1} \sin\frac{\pi k}{n}=\cot\frac{\pi}{2n}\,\!
  • \sum_{k=1}^{n-1} \sin\frac{2\pi k}{n}=0\,\!
  • \sum_{k=1}^{n-1} \csc^2\frac{\pi k}{n}=\frac{n^2-1}{3}\,\!
  • \sum_{k=1}^{n-1} \csc^4\frac{\pi k}{n}=\frac{n^4+10n^2-11}{45}\,\!

Rational functions[edit]

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

See also[edit]

Notes[edit]

References[edit]