List of mathematical series

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

• Here, [math]\displaystyle{ 0^0 }[/math] is taken to have the value [math]\displaystyle{ 1 }[/math]
• [math]\displaystyle{ \{x\} }[/math] denotes the fractional part of [math]\displaystyle{ x }[/math]
• [math]\displaystyle{ B_n(x) }[/math] is a Bernoulli polynomial.
• [math]\displaystyle{ B_n }[/math] is a Bernoulli number, and here, [math]\displaystyle{ B_1=-\frac{1}{2}. }[/math]
• [math]\displaystyle{ E_n }[/math] is an Euler number.
• [math]\displaystyle{ \zeta(s) }[/math] is the Riemann zeta function.
• [math]\displaystyle{ \Gamma(z) }[/math] is the gamma function.
• [math]\displaystyle{ \psi_n(z) }[/math] is a polygamma function.
• [math]\displaystyle{ \operatorname{Li}_s(z) }[/math] is a polylogarithm.
• [math]\displaystyle{ n \choose k }[/math] is binomial coefficient
• [math]\displaystyle{ \exp(x) }[/math] denotes exponential of [math]\displaystyle{ x }[/math]

Sums of powers

See Faulhaber's formula.

• [math]\displaystyle{ \sum_{k=0}^m k^{n-1}=\frac{B_n(m+1)-B_n}{n} }[/math]

The first few values are:

• [math]\displaystyle{ \sum_{k=1}^m k=\frac{m(m+1)}{2} }[/math]
• [math]\displaystyle{ \sum_{k=1}^m k^2=\frac{m(m+1)(2m+1)}{6}=\frac{m^3}{3}+\frac{m^2}{2}+\frac{m}{6} }[/math]
• [math]\displaystyle{ \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} }[/math]

See zeta constants.

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

The first few values are:

• [math]\displaystyle{ \zeta(2)=\sum^{\infty}_{k=1} \frac{1}{k^2}=\frac{\pi^2}{6} }[/math] (the Basel problem)
• [math]\displaystyle{ \zeta(4)=\sum^{\infty}_{k=1} \frac{1}{k^4}=\frac{\pi^4}{90} }[/math]
• [math]\displaystyle{ \zeta(6)=\sum^{\infty}_{k=1} \frac{1}{k^6}=\frac{\pi^6}{945} }[/math]

Power series

Low-order polylogarithms

Finite sums:

• [math]\displaystyle{ \sum_{k=m}^{n} z^k = \frac{z^{m}-z^{n+1}}{1-z} }[/math], (geometric series)
• [math]\displaystyle{ \sum_{k=0}^{n} z^k = \frac{1-z^{n+1}}{1-z} }[/math]
• [math]\displaystyle{ \sum_{k=1}^{n} z^k = \frac{1-z^{n+1}}{1-z}-1 = \frac{z-z^{n+1}}{1-z} }[/math]
• [math]\displaystyle{ \sum_{k=1}^n k z^k = z\frac{1-(n+1)z^n+nz^{n+1}}{(1-z)^2} }[/math]
• [math]\displaystyle{ \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} }[/math]
• [math]\displaystyle{ \sum_{k=1}^n k^m z^k = \left(z \frac{d}{dz}\right)^m \frac{1-z^{n+1}}{1-z} }[/math]

Infinite sums, valid for [math]\displaystyle{ |z|\lt 1 }[/math] (see polylogarithm):

• [math]\displaystyle{ \operatorname{Li}_n(z)=\sum_{k=1}^{\infty} \frac{z^k}{k^n} }[/math]

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

• [math]\displaystyle{ \frac{\mathrm{d}}{\mathrm{d}z}\operatorname{Li}_n(z)=\frac{\operatorname{Li}_{n-1}(z)}{z} }[/math]
• [math]\displaystyle{ \operatorname{Li}_{1}(z)=\sum_{k=1}^\infty \frac{z^k}{k}=-\ln(1-z) }[/math]
• [math]\displaystyle{ \operatorname{Li}_{0}(z)=\sum_{k=1}^\infty z^k=\frac{z}{1-z} }[/math]
• [math]\displaystyle{ \operatorname{Li}_{-1}(z)=\sum_{k=1}^\infty k z^k=\frac{z}{(1-z)^2} }[/math]
• [math]\displaystyle{ \operatorname{Li}_{-2}(z)=\sum_{k=1}^\infty k^2 z^k=\frac{z(1+z)}{(1-z)^3} }[/math]
• [math]\displaystyle{ \operatorname{Li}_{-3}(z)=\sum_{k=1}^\infty k^3 z^k =\frac{z(1+4z+z^2)}{(1-z)^4} }[/math]
• [math]\displaystyle{ \operatorname{Li}_{-4}(z)=\sum_{k=1}^\infty k^4 z^k =\frac{z(1+z)(1+10z+z^2)}{(1-z)^5} }[/math]

Exponential function

• [math]\displaystyle{ \sum_{k=0}^\infty \frac{z^k}{k!} = e^z }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty k\frac{z^k}{k!} = z e^z }[/math] (cf. mean of Poisson distribution)
• [math]\displaystyle{ \sum_{k=0}^\infty k^2 \frac{z^k}{k!} = (z + z^2) e^z }[/math] (cf. second moment of Poisson distribution)
• [math]\displaystyle{ \sum_{k=0}^\infty k^3 \frac{z^k}{k!} = (z + 3z^2 + z^3) e^z }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty k^4 \frac{z^k}{k!} = (z + 7z^2 + 6z^3 + z^4) e^z }[/math]
• [math]\displaystyle{ \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) }[/math]

where [math]\displaystyle{ T_{n}(z) }[/math] is the Touchard polynomials.

Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship

• [math]\displaystyle{ \sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)!}=\sin z }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{z^{2k+1}}{(2k+1)!}=\sinh z }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{(2k)!}=\cos z }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{z^{2k}}{(2k)!}=\cosh z }[/math]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{(-1)^{k-1}(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\tan z, |z|\lt \frac{\pi}{2} }[/math]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\tanh z, |z|\lt \frac{\pi}{2} }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{(-1)^k2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\cot z, |z|\lt \pi }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\coth z, |z|\lt \pi }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{(-1)^{k-1}(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\csc z, |z|\lt \pi }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{-(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\operatorname{csch} z, |z|\lt \pi }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{(-1)^kE_{2k}z^{2k}}{(2k)!}=\operatorname{sech} z, |z|\lt \frac{\pi}{2} }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{E_{2k}z^{2k}}{(2k)!}=\sec z, |z| \lt \frac{\pi}{2} }[/math]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{(-1)^{k-1} z^{2k}}{(2k)!}=\operatorname{ver}z }[/math] (versine)
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{(-1)^{k-1} z^{2k}}{2(2k)!}=\operatorname{hav}z }[/math]^[1] (haversine)
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{(2k)!z^{2k+1}}{2^{2k}(k!)^2(2k+1)}=\arcsin z, |z|\le1 }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{(-1)^k(2k)!z^{2k+1}}{2^{2k}(k!)^2(2k+1)}=\operatorname{arcsinh} {z}, |z| \le 1 }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{(-1)^kz^{2k+1}}{2k+1}=\arctan z, |z|\lt 1 }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{z^{2k+1}}{2k+1}=\operatorname{arctanh} z, |z|\lt 1 }[/math]
• [math]\displaystyle{ \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| \le 1 }[/math]
• [math]\displaystyle{ \sum_{k=2}^\infty \left( k \cdot \operatorname{arctanh}\left(\frac{1}{k}\right) - 1 \right) = \frac{3-\ln(4 \pi)}{2} }[/math]

Modified-factorial denominators

• [math]\displaystyle{ \sum^{\infty}_{k=0} \frac{(4k)!}{2^{4k} \sqrt{2} (2k)! (2k+1)!} z^k = \sqrt{\frac{1-\sqrt{1-z}}{z}}, |z|\lt 1 }[/math]^[2]
• [math]\displaystyle{ \sum^{\infty}_{k=0} \frac{2^{2k} (k!)^2}{(k+1) (2k+1)!} z^{2k+2} = \left(\arcsin{z}\right)^2, |z|\le1 }[/math]^[2]
• [math]\displaystyle{ \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 }[/math]

Binomial coefficients

• [math]\displaystyle{ (1+z)^\alpha = \sum_{k=0}^\infty {\alpha \choose k} z^k, |z|\lt 1 }[/math] (see Binomial theorem § Newton's generalized binomial theorem)
• ^[3] [math]\displaystyle{ \sum_{k=0}^\infty {{\alpha+k-1} \choose k} z^k = \frac{1}{(1-z)^\alpha}, |z|\lt 1 }[/math]
• ^[3] [math]\displaystyle{ \sum_{k=0}^\infty \frac{1}{k+1}{2k \choose k} z^k = \frac{1-\sqrt{1-4z}}{2z}, |z|\leq\frac{1}{4} }[/math], generating function of the Catalan numbers
• ^[3] [math]\displaystyle{ \sum_{k=0}^\infty {2k \choose k} z^k = \frac{1}{\sqrt{1-4z}}, |z|\lt \frac{1}{4} }[/math], generating function of the Central binomial coefficients
• ^[3] [math]\displaystyle{ \sum_{k=0}^\infty {2k + \alpha \choose k} z^k = \frac{1}{\sqrt{1-4z}}\left(\frac{1-\sqrt{1-4z}}{2z}\right)^\alpha, |z|\lt \frac{1}{4} }[/math]

Harmonic numbers

(See harmonic numbers, themselves defined [math]\displaystyle{ H_n = \sum_{j=1}^{n} \frac{1}{j} }[/math], and [math]\displaystyle{ H(x) }[/math] generalized to the real numbers)

• [math]\displaystyle{ \sum_{k=1}^\infty H_k z^k = \frac{-\ln(1-z)}{1-z}, |z|\lt 1 }[/math]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{H_k}{k+1} z^{k+1} = \frac{1}{2}\left[\ln(1-z)\right]^2, \qquad |z|\lt 1 }[/math]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{(-1)^{k-1} H_{2k}}{2k+1} z^{2k+1} = \frac{1}{2} \arctan{z} \log{(1+z^2)}, \qquad |z|\lt 1 }[/math]^[2]
• [math]\displaystyle{ \sum_{n=0}^\infty \sum_{k=0}^{2n} \frac{(-1)^k}{2k+1} \frac{z^{4n+2}}{4n+2} = \frac{1}{4} \arctan{z} \log{\frac{1+z}{1-z}},\qquad |z|\lt 1 }[/math]^[2]
• [math]\displaystyle{ \sum_{n=0}^\infty \frac{x^2}{n^2(n+x)} = x\frac{\pi^2}{6} - H(x) }[/math]

Binomial coefficients

Main page: Binomial coefficient

• [math]\displaystyle{ \sum_{k=0}^n {n \choose k} = 2^n }[/math]
• [math]\displaystyle{ \sum_{k=0}^n {n \choose k}^2 = {2n \choose n} }[/math]
• [math]\displaystyle{ \sum_{k=0}^n (-1)^k {n \choose k} = 0, \text{ where }n\geq 1 }[/math]
• [math]\displaystyle{ \sum_{k=0}^n {k \choose m} = { n+1 \choose m+1 } }[/math]
• [math]\displaystyle{ \sum_{k=0}^n {m+k-1 \choose k} = { n+m \choose n } }[/math] (see Multiset)
• [math]\displaystyle{ \sum_{k=0}^n {\alpha \choose k}{\beta \choose n-k} = {\alpha+\beta \choose n}, \text{where} \ \alpha + \beta \geq n }[/math] (see Vandermonde identity)
• [math]\displaystyle{ \sum_{A \ \in \ \mathcal{P}(E)} 1 = 2^n \text{, where }E\text{ is a finite set, and card(}E\text{) = n} }[/math]
• [math]\displaystyle{ \sum_{\begin{cases} (A,\ B) \ \in \ (\mathcal{P}(E))^2 \\ A \ \subset\ B \end{cases}} 1 = 3^n\text{, where }E\text{ is a finite set, and card(}E\text{) = n} }[/math]
• [math]\displaystyle{ \sum_{A \ \in \ \mathcal{P}(E)} card(A) = n2^{n-1} \text{, where }E\text{ is a finite set, and card(}E\text{) = n} }[/math]

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

• [math]\displaystyle{ \sum_{k=1}^\infty \frac{\cos(k\theta)}{k}=-\frac{1}{2}\ln(2-2\cos\theta)=-\ln \left(2\sin\frac{\theta}{2} \right), 0\lt \theta\lt 2\pi }[/math]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{\sin(k\theta)}{k}=\frac{\pi-\theta}{2}, 0\lt \theta\lt 2\pi }[/math]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\cos(k\theta)=\frac{1}{2}\ln(2+2\cos\theta)=\ln \left(2\cos\frac{\theta}{2}\right), 0\leq\theta\lt \pi }[/math]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\sin(k\theta)=\frac{\theta}{2}, -\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2} }[/math]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{\cos(2k\theta)}{2k}=-\frac{1}{2}\ln(2\sin\theta), 0\lt \theta\lt \pi }[/math]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{\sin(2k\theta)}{2k}=\frac{\pi-2\theta}{4}, 0\lt \theta\lt \pi }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{\cos[(2k+1)\theta]}{2k+1}=\frac{1}{2}\ln \left(\cot\frac{\theta}{2}\right), 0\lt \theta\lt \pi }[/math]
• [math]\displaystyle{ \sum_{k=0}^\infty \frac{\sin[(2k+1)\theta]}{2k+1}=\frac{\pi}{4}, 0\lt \theta\lt \pi }[/math],^[4]
• [math]\displaystyle{ \sum_{k=1}^\infty \frac{\sin(2 \pi k x)}{k}= \pi \left(\dfrac{1}{2} - \{x\}\right), \ x \in \mathbb{R} }[/math]
• [math]\displaystyle{ \sum\limits_{k=1}^{\infty} \frac{\sin \left(2\pi kx \right)}{k^{2n-1}} = (-1)^{n}\frac{(2\pi)^{2n-1}}{2(2n-1)!} B_{2n-1}(\{x\}), \ x \in \mathbb{R}, \ n \in \mathbb{N} }[/math]
• [math]\displaystyle{ \sum\limits_{k=1}^{\infty} \frac{\cos \left(2\pi kx \right)}{k^{2n}} = (-1)^{n-1}\frac{(2\pi)^{2n}}{2(2n)!} B_{2n}(\{x\}), \ x \in \mathbb{R}, \ n \in \mathbb{N} }[/math]
• [math]\displaystyle{ B_n(x)=-\frac{n!}{2^{n-1}\pi^n}\sum_{k=1}^\infty \frac{1}{k^n}\cos\left(2\pi kx-\frac{\pi n}{2}\right), 0\lt x\lt 1 }[/math]^[5]
• [math]\displaystyle{ \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}} }[/math]
• [math]\displaystyle{ \sum_{k=0}^n \cos(\theta+k\alpha)=\frac{\sin\frac{(n+1)\alpha}{2}\cos(\theta+\frac{n\alpha}{2})}{\sin\frac{\alpha}{2}} }[/math]
• [math]\displaystyle{ \sum_{k=1}^{n-1} \sin\frac{\pi k}{n}=\cot\frac{\pi}{2n} }[/math]
• [math]\displaystyle{ \sum_{k=1}^{n-1} \sin\frac{2\pi k}{n}=0 }[/math]
• [math]\displaystyle{ \sum_{k=0}^{n-1} \csc^2\left(\theta+\frac{\pi k}{n}\right)=n^2\csc^2(n\theta) }[/math]^[6]
• [math]\displaystyle{ \sum_{k=1}^{n-1} \csc^2\frac{\pi k}{n}=\frac{n^2-1}{3} }[/math]
• [math]\displaystyle{ \sum_{k=1}^{n-1} \csc^4\frac{\pi k}{n}=\frac{n^4+10n^2-11}{45} }[/math]

Rational functions

• [math]\displaystyle{ \sum_{n=a+1}^{\infty} \frac{a}{n^2 - a^2} = \frac{1}{2} H_{2a} }[/math]^[7]
• [math]\displaystyle{ \sum_{n=0}^\infty\frac{1}{n^2+a^2}=\frac{1+a\pi\coth (a\pi)}{2a^2} }[/math]
• [math]\displaystyle{ \sum_{n=0}^\infty\frac{(-1)^n}{n^2+a^2} = \frac{1 + a\pi \; \text{csch}(a\pi)}{2a^2} }[/math]
• [math]\displaystyle{ \sum_{n=0}^\infty\frac{(2n+1)(-1)^n}{(2n+1)^2+a^2}= \frac{\pi}{4} \text{sech} \left( \frac{a \pi}{2} \right) }[/math]
• [math]\displaystyle{ \displaystyle \sum_{n=0}^\infty \frac {1}{n^4+4a^4} = \dfrac{1}{8a^4}+\dfrac{\pi(\sinh(2\pi a)+\sin(2\pi a))}{8a^3(\cosh(2\pi a)-\cos(2\pi a))} }[/math]
• An infinite series of any rational function of [math]\displaystyle{ n }[/math] can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition,^[8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.

Exponential function

• [math]\displaystyle{ \displaystyle \dfrac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp \left(\frac{2\pi i n^2 q}{p} \right)=\dfrac{e^{\pi i/4}}{\sqrt{2q}}\sum_{n=0}^{2q-1}\exp \left(-\frac{\pi i n^2 p}{2q} \right) }[/math](see the Landsberg–Schaar relation)
• [math]\displaystyle{ \displaystyle \sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)} }[/math]

Numeric series

These numeric series can be found by plugging in numbers from the series listed above.

Alternating harmonic series

• [math]\displaystyle{ \sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{k}=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots=\ln 2 }[/math]
• [math]\displaystyle{ \sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{2k-1}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots=\frac{\pi}{4} }[/math]

Sum of reciprocal of factorials

• [math]\displaystyle{ \sum^{\infty}_{k=0} \frac{1}{k!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots=e }[/math]
• [math]\displaystyle{ \sum^{\infty}_{k=0} \frac{1}{(2k)!}=\frac{1}{0!}+\frac{1}{2!}+\frac{1}{4!}+\frac{1}{6!}+\frac{1}{8!}+\cdots=\frac{1}{2}\left(e+\frac{1}{e}\right)=\cosh 1 }[/math]
• [math]\displaystyle{ \sum^{\infty}_{k=0} \frac{1}{(3k)!}=\frac{1}{0!}+\frac{1}{3!}+\frac{1}{6!}+\frac{1}{9!}+\frac{1}{12!}+\cdots=\frac{1}{3}\left(e+\frac{2}{\sqrt{e}}\cos \frac{\sqrt{3}}{2}\right) }[/math]
• [math]\displaystyle{ \sum^{\infty}_{k=0} \frac{1}{(4k)!}=\frac{1}{0!}+\frac{1}{4!}+\frac{1}{8!}+\frac{1}{12!}+\frac{1}{16!}+\cdots=\frac{1}{2}\left(\cos 1+\cosh 1\right) }[/math]

Trigonometry and π

• [math]\displaystyle{ \sum^{\infty}_{k=0} \frac{(-1)^k}{(2k+1)!}=\frac{1}{1!}-\frac{1}{3!}+\frac{1}{5!}-\frac{1}{7!}+\frac{1}{9!}+\cdots=\sin 1 }[/math]
• [math]\displaystyle{ \sum^{\infty}_{k=0} \frac{(-1)^k}{(2k)!}=\frac{1}{0!}-\frac{1}{2!}+\frac{1}{4!}-\frac{1}{6!}+\frac{1}{8!}+\cdots=\cos 1 }[/math]
• [math]\displaystyle{ \sum^{\infty}_{k=1} \frac{1}{k^2+1}=\frac{1}{2}+\frac{1}{5}+\frac{1}{10}+\frac{1}{17}+\cdots=\frac{1}{2}(\pi \coth \pi - 1) }[/math]
• [math]\displaystyle{ \sum^{\infty}_{k=1} \frac{(-1)^k}{k^2+1}=-\frac{1}{2}+\frac{1}{5}-\frac{1}{10}+\frac{1}{17}+\cdots=\frac{1}{2}(\pi \operatorname{csch} \pi - 1) }[/math]
• [math]\displaystyle{ 3 + \frac{4}{2\times3\times4} - \frac{4}{4\times5\times6} + \frac{4}{6\times7\times8} - \frac{4}{8\times9\times10} + \cdots = \pi }[/math]

Reciprocal of triangular numbers

• [math]\displaystyle{ \sum^\infty_{k=1} \frac{1}{T_k}=\frac{1}{1}+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+\frac{1}{15}+\cdots=2 }[/math]

Where [math]\displaystyle{ T_n=\sum^n_{k=1} k }[/math]

Reciprocal of tetrahedral numbers

• [math]\displaystyle{ \sum^{\infty}_{k=1} \frac{1}{Te_k}=\frac{1}{1}+\frac{1}{4}+\frac{1}{10}+\frac{1}{20}+\frac{1}{35}+\cdots=\frac{3}{2} }[/math]

Where [math]\displaystyle{ Te_n=\sum^{n}_{k=1} T_k }[/math]

Exponential and logarithms

• [math]\displaystyle{ \sum^{\infty}_{k=0} \frac{1}{(2k+1)(2k+2)}=\frac{1}{1\times 2}+\frac{1}{3\times 4}+\frac{1}{5\times 6}+\frac{1}{7\times 8}+\frac{1}{9\times 10}+\cdots=\ln 2 }[/math]
• [math]\displaystyle{ \sum^{\infty}_{k=1} \frac{1}{2^kk}=\frac{1}{2}+\frac{1}{8}+\frac{1}{24}+\frac{1}{64}+\frac{1}{160}+\cdots=\ln 2 }[/math]
• [math]\displaystyle{ \sum^{\infty}_{k=1} \frac{(-1)^{k+1}}{2^kk}+\sum^{\infty}_{k=1} \frac{(-1)^{k+1}}{3^kk}=\Bigg(\frac{1}{2}+\frac{1}{3}\Bigg)-\Bigg(\frac{1}{8}+\frac{1}{18}\Bigg)+\Bigg(\frac{1}{24}+\frac{1}{81}\Bigg)-\Bigg(\frac{1}{64}+\frac{1}{324}\Bigg)+\cdots=\ln 2 }[/math]
• [math]\displaystyle{ \sum^{\infty}_{k=1} \frac{1}{3^kk}+\sum^{\infty}_{k=1} \frac{1}{4^kk}=\Bigg(\frac{1}{3}+\frac{1}{4}\Bigg)+\Bigg(\frac{1}{18}+\frac{1}{32}\Bigg)+\Bigg(\frac{1}{81}+\frac{1}{192}\Bigg)+\Bigg(\frac{1}{324}+\frac{1}{1024}\Bigg)+\cdots=\ln 2 }[/math]

See also

• Series (mathematics)
• List of integrals
• Summation § Identities
• Taylor series
• Binomial theorem
• Gregory's series
• On-Line Encyclopedia of Integer Sequences

Notes

References

• Many books with a list of integrals also have a list of series.
• [math]\displaystyle{ \frac{\pi}{4} = \sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{2k+1} }[/math]

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