List of mathematical series

Short description: none

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, 00 is taken to have the value 1
* {x} denotes the fractional part of x
* Bn(x) is a Bernoulli polynomial.
* Bn is a Bernoulli number, and here, B1=−12.
* En is an Euler number.
* ζ(s) is the Riemann zeta function.
* Γ(z) is the gamma function.
* ψn(z) is a polygamma function.
* Lis(z) is a polylogarithm.
* (nk) is binomial coefficient
* exp⁡(x) denotes exponential of x

## Contents

* 1 Sums of powers
* 2 Power series
* 2.1 Low-order polylogarithms
* 2.2 Exponential function
* 2.3 Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship
* 2.4 Modified-factorial denominators
* 2.5 Binomial coefficients
* 2.6 Harmonic numbers
* 3 Binomial coefficients
* 4 Trigonometric functions
* 5 Roots of unity
* 6 Rational functions
* 7 Exponential function
* 8 Numeric series
* 8.1 Alternating harmonic series
* 8.2 Alternating arithmetic series
* 8.3 Sum of reciprocal of factorials
* 8.4 Trigonometry and π
* 8.5 Reciprocal of tetrahedral numbers
* 8.6 Exponential and logarithms
* 9 See also
* 10 Notes
* 11 References

## Sums of powers

See Faulhaber's formula.

* ∑k=0mkn−1=Bn(m+1)−Bnn

The first few values are:

* ∑k=1mk=m(m+1)2
* ∑k=1mk2=m(m+1)(2m+1)6=m33+m22+m6
* ∑k=1mk3=[m(m+1)2]2=m44+m32+m24

See zeta constants.

* ζ(2n)=∑k=1∞1k2n=(−1)n+1B2n(2π)2n2(2n)!

The first few values are:

* ζ(2)=∑k=1∞1k2=π26 (the Basel problem)
* ζ(4)=∑k=1∞1k4=π490
* ζ(6)=∑k=1∞1k6=π6945

## Power series

### Low-order polylogarithms

Finite sums:

* ∑k=mnzk=zm−zn+11−z (geometric series)
* ∑k=0nzk=1−zn+11−z
* ∑k=1nzk=1−zn+11−z−1=z−zn+11−z
* ∑k=1nkzk=z1−(n+1)zn+nzn+1(1−z)2
* ∑k=1nk2zk=z1+z−(n+1)2zn+(2n2+2n−1)zn+1−n2zn+2(1−z)3
* ∑k=0nkmzk=(zddz)m1−zn+11−z

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

* Lin(z)=∑k=1∞zkkn

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

* ddzLin(z)=Lin−1(z)z
* Li1(z)=∑k=1∞zkk=−ln⁡(1−z)
* Li0(z)=∑k=1∞zk=z1−z
* Li−1(z)=∑k=1∞kzk=z(1−z)2
* Li−2(z)=∑k=1∞k2zk=z(1+z)(1−z)3
* Li−3(z)=∑k=1∞k3zk=z(1+4z+z2)(1−z)4
* Li−4(z)=∑k=1∞k4zk=z(1+z)(1+10z+z2)(1−z)5

### Exponential function

* ∑k=0∞zkk!=ez
* ∑k=0∞kzkk!=zez (cf. mean of Poisson distribution)
* ∑k=0∞k2zkk!=(z+z2)ez (cf. second moment of Poisson distribution)
* ∑k=0∞k3zkk!=(z+3z2+z3)ez
* ∑k=0∞k4zkk!=(z+7z2+6z3+z4)ez
* ∑k=0∞knzkk!=zddz∑k=0∞kn−1zkk!=ezTn(z)

where Tn(z) is the Touchard polynomials.

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

* ∑k=0∞(−1)kz2k+1(2k+1)!=sin⁡z
* ∑k=0∞z2k+1(2k+1)!=sinh⁡z
* ∑k=0∞(−1)kz2k(2k)!=cos⁡z
* ∑k=0∞z2k(2k)!=cosh⁡z
* ∑k=1∞(−1)k−1(22k−1)22kB2kz2k−1(2k)!=tan⁡z,|z|<π2
* ∑k=1∞(22k−1)22kB2kz2k−1(2k)!=tanh⁡z,|z|<π2
* ∑k=0∞(−1)k22kB2kz2k−1(2k)!=cot⁡z,|z|<π
* ∑k=0∞22kB2kz2k−1(2k)!=coth⁡z,|z|<π
* ∑k=0∞(−1)k−1(22k−2)B2kz2k−1(2k)!=csc⁡z,|z|<π
* ∑k=0∞−(22k−2)B2kz2k−1(2k)!=csch⁡z,|z|<π
* ∑k=0∞(−1)kE2kz2k(2k)!=sech⁡z,|z|<π2
* ∑k=0∞E2kz2k(2k)!=sec⁡z,|z|<π2
* ∑k=1∞(−1)k−1z2k(2k)!=ver⁡z (versine)
* ∑k=1∞(−1)k−1z2k2(2k)!=hav⁡z[1] (haversine)
* ∑k=0∞(2k)!z2k+122k(k!)2(2k+1)=arcsin⁡z,|z|≤1
* ∑k=0∞(−1)k(2k)!z2k+122k(k!)2(2k+1)=arcsinh⁡z,|z|≤1
* ∑k=0∞(−1)kz2k+12k+1=arctan⁡z,|z|<1
* ∑k=0∞z2k+12k+1=arctanh⁡z,|z|<1
* ln⁡2+∑k=1∞(−1)k−1(2k)!z2k22k+1k(k!)2=ln⁡(1+1+z2),|z|≤1
* ∑k=2∞(k⋅arctanh⁡(1k)−1)=3−ln⁡(4π)2

### Modified-factorial denominators

* ∑k=0∞(4k)!24k2(2k)!(2k+1)!zk=1−1−zz,|z|<1[2]
* ∑k=0∞22k(k!)2(k+1)(2k+1)!z2k+2=(arcsin⁡z)2,|z|≤1[2]
* ∑n=0∞∏k=0n−1(4k2+α2)(2n)!z2n+∑n=0∞α∏k=0n−1[(2k+1)2+α2](2n+1)!z2n+1=eαarcsin⁡z,|z|≤1

### Binomial coefficients

* (1+z)α=∑k=0∞(αk)zk,|z|<1 (see Binomial theorem § Newton's generalized binomial theorem)
* [3] ∑k=0∞(α+k−1k)zk=1(1−z)α,|z|<1
* [3] ∑k=0∞1k+1(2kk)zk=1−1−4z2z,|z|≤14, generating function of the Catalan numbers
* [3] ∑k=0∞(2kk)zk=11−4z,|z|<14, generating function of the Central binomial coefficients
* [3] ∑k=0∞(2k+αk)zk=11−4z(1−1−4z2z)α,|z|<14
* ∑k=0∞1(N+kn)=N(n−1)(Nn),N≥n

### Harmonic numbers

(See harmonic numbers, themselves defined Hn=∑j=1n1j, and H(x) generalized to the real numbers)

* ∑k=1∞Hkzk=−ln⁡(1−z)1−z,|z|<1
* ∑k=1∞Hkk+1zk+1=12[ln⁡(1−z)]2,|z|<1
* ∑k=1∞(−1)k−1H2k2k+1z2k+1=12arctan⁡zlog⁡(1+z2),|z|<1[2]
* ∑n=0∞∑k=02n(−1)k2k+1z4n+24n+2=14arctan⁡zlog⁡1+z1−z,|z|<1[2]
* ∑n=1∞x2n2(n+x)=xπ26−H(x)

## Binomial coefficients

Main article: Binomial coefficient

* ∑k=0n(nk)=2n
* ∑k=0n(nk)2=(2nn)
* ∑k=0n(−1)k(nk)=0, where n≥1
* ∑k=0n(km)=(n+1m+1)
* ∑k=0n(m+k−1k)=(n+mn) (see Multiset)
* ∑k=0n(αk)(βn−k)=(α+βn),where α+β≥n (see Vandermonde identity)
* ∑A ∈ 𝒫(E)1=2n, where E is a finite set, and card(E) = n
* ∑{(A, B) ∈ (𝒫(E))2A ⊂ B1=3n, where E is a finite set, and card(E) = n
* ∑A ∈ 𝒫(E)card(A)=n2n−1, where E is a finite set, and card(E) = n

## Trigonometric functions

Sums of sines and cosines arise in Fourier series.

* ∑k=1∞cos⁡(kθ)k=−12ln⁡(2−2cos⁡θ)=−ln⁡(2sin⁡θ2),0<θ<2π
* ∑k=1∞sin⁡(kθ)k=π−θ2,0<θ<2π
* ∑k=1∞(−1)k−1kcos⁡(kθ)=12ln⁡(2+2cos⁡θ)=ln⁡(2cos⁡θ2),0≤θ<π
* ∑k=1∞(−1)k−1ksin⁡(kθ)=θ2,−π2≤θ≤π2
* ∑k=1∞cos⁡(2kθ)2k=−12ln⁡(2sin⁡θ),0<θ<π
* ∑k=1∞sin⁡(2kθ)2k=π−2θ4,0<θ<π
* ∑k=0∞cos⁡[(2k+1)θ]2k+1=12ln⁡(cot⁡θ2),0<θ<π
* ∑k=0∞sin⁡[(2k+1)θ]2k+1=π4,0<θ<π,[4]
* ∑k=1∞sin⁡(2πkx)k=π(12−{x}), x∈ℝ
* ∑k=1∞sin⁡(2πkx)k2n−1=(−1)n(2π)2n−12(2n−1)!B2n−1({x}), x∈ℝ, n∈ℕ
* ∑k=1∞cos⁡(2πkx)k2n=(−1)n−1(2π)2n2(2n)!B2n({x}), x∈ℝ, n∈ℕ
* Bn(x)=−n!2n−1πn∑k=1∞1kncos⁡(2πkx−πn2),0<x<1[5]
* ∑k=0nsin⁡(θ+kα)=sin⁡(n+1)α2sin⁡(θ+nα2)sin⁡α2
* ∑k=0ncos⁡(θ+kα)=sin⁡(n+1)α2cos⁡(θ+nα2)sin⁡α2
* ∑k=1n−1sin⁡πkn=cot⁡π2n
* ∑k=1n−1sin⁡2πkn=0
* ∑k=0n−1csc2(θ+πkn)=n2csc2(nθ)[6]
* ∑k=1n−1csc2πkn=n2−13
* ∑k=1n−1csc4πkn=n4+10n2−1145

## Roots of unity

A n'th root of unity is a solution to the equation zn=1 and they can be written like:

zm=exp⁡(i⋅2πmn),m∈{0,⋯,n−1}

The following summation identities hold:

∑m≠0n−111−zm=n−12

∑m≠0n−11(1−zm)2=(n−1)(5−n)12

Let k be an integer 0<k<n then we also got:

∑m=0n−1zmk=0

∑m=0n−1(x−zm)k=n⋅xk

∑m≠0n−1(zm)k1−zm=2k−n−12

∑m≠0n−1(zm)k(1−zm)2=6k(n+2−k)−(n+1)(n+5)12

## Rational functions

* ∑n=a+1∞an2−a2=12H2a[7]
* ∑n=0∞1n2+a2=1+aπcoth⁡(aπ)2a2
* ∑n=0∞(−1)nn2+a2=1+aπcsch(aπ)2a2
* ∑n=0∞(2n+1)(−1)n(2n+1)2+a2=π4sech(aπ2)
* ∑n=0∞1n4+4a4=18a4+π(sinh⁡(2πa)+sin⁡(2πa))8a3(cosh⁡(2πa)−cos⁡(2πa))
* An infinite series of any rational function of n 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

* 1p∑n=0p−1exp⁡(2πin2qp)=eπi/42q∑n=02q−1exp⁡(−πin2p2q)(see the Landsberg–Schaar relation)
* ∑n=−∞∞e−πn2=π4Γ(34)

## Numeric series

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

### Alternating harmonic series

* ∑k=1∞(−1)k+1k=11−12+13−14+⋯=ln⁡2
* ∑k=1∞(−1)k+12k−1=11−13+15−17+19−⋯=π4

### Alternating arithmetic series

Let S(a,b) be defined as:

S(a,b)=∑k=0∞(−1)kak+b

where a,b>0 are positive whole numbers. Then if gcd(a,b)=c we can write a=cα and b=cβ, where gcd(α,β)=1, and get:

S(a,b)=S(cα,cβ)=∑k=0∞(−1)kcdk+ce=1c∑k=0∞(−1)kαk+β=S(α,β)c

Now if b>a we can, per Euclid's division lemma, write b=ca+d where a>d>0 and then

S(a,b)=S(a,ca+d)=∑k=0∞(−1)kak+ca+d=∑k=0∞(−1)ka(k+c)+d=∑k=c∞(−1)k−cak+d

where we now can add the remaining rows back and subtract them to give us:

S(a,b)=(−1)c(∑k=0∞(−1)kak+d−∑k=0c−1(−1)kak+d)=(−1)c(S(a,d)−∑k=0c−1(−1)kak+d)

what that means is that all the infinite choices of a and b can essentially be boiled down to the cases where gcd(a,b)=1 and a>b>0. If we assume those two things we can then write:

S(a,b)=1a(π2sin⁡(πba)−2∑m=0⌊a2⌋cos⁡(π(2m+1)ba)ln⁡(sin⁡(π2m+12a)))

and in the case of using a negative sign instead:

S−(a,b)=∑k=0∞(−1)kak−b

the same two rules apply from above apply and then we can do the following for the case with a>b>0 (since a>a−b>0):

S−(a,b)=∑k=0∞(−1)kak−b=−1b+∑k=0∞(−1)k+1a(k+1)−b=−1b−∑k=0∞(−1)kak+(a−b)=−S(a,a−b)−1b

Let us test out the formula: S(3,2)=13(π2sin⁡(2π3)−2(cos⁡(2π3)ln⁡(sin⁡(π6))+cos⁡(2π)ln⁡(sin⁡(π2))))=π33−ln⁡(2)3

### Sum of reciprocal of factorials

* ∑k=0∞1k!=10!+11!+12!+13!+14!+⋯=e
* ∑k=0∞1(2k)!=10!+12!+14!+16!+18!+⋯=12(e+1e)=cosh⁡1
* ∑k=0∞1(3k)!=10!+13!+16!+19!+112!+⋯=13(e+2ecos⁡32)
* ∑k=0∞1(4k)!=10!+14!+18!+112!+116!+⋯=12(cos⁡1+cosh⁡1)

### Trigonometry and π

* ∑k=0∞(−1)k(2k+1)!=11!−13!+15!−17!+19!+⋯=sin⁡1
* ∑k=0∞(−1)k(2k)!=10!−12!+14!−16!+18!+⋯=cos⁡1
* ∑k=1∞1k2+1=12+15+110+117+⋯=12(πcoth⁡π−1)
* ∑k=1∞(−1)kk2+1=−12+15−110+117+⋯=12(πcsch⁡π−1)
* 3+42×3×4−44×5×6+46×7×8−48×9×10+⋯=π

### Reciprocal of tetrahedral numbers

* ∑k=1∞1Tek=11+14+110+120+135+⋯=32

Where Ten=∑k=1nTk

### Exponential and logarithms

* ∑k=0∞1(2k+1)(2k+2)=11×2+13×4+15×6+17×8+19×10+⋯=ln⁡2
* ∑k=1∞12kk=12+18+124+164+1160+⋯=ln⁡2
* ∑k=1∞(−1)k+12kk+∑k=1∞(−1)k+13kk=(12+13)−(18+118)+(124+181)−(164+1324)+⋯=ln⁡2
* ∑k=1∞13kk+∑k=1∞14kk=(13+14)+(118+132)+(181+1192)+(1324+11024)+⋯=ln⁡2
* ∑k=1∞1nkk=ln⁡(nn−1), that is ∀n>1

## See also

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

## Notes

1. ↑ Weisstein, Eric W.. "Haversine". MathWorld. Wolfram Research, Inc.. http://mathworld.wolfram.com/Haversine.html.
2. ↑ 2.0 2.1 2.2 2.3 Wilf, Herbert R. (1994). generatingfunctionology. Academic Press, Inc. http://www.math.upenn.edu/~wilf/gfologyLinked2.pdf.
3. ↑ 3.0 3.1 3.2 3.3 "Theoretical computer science cheat sheet". http://www.tug.org/texshowcase/cheat.pdf.
4. ↑ Calculate the Fourier expansion of the function f(x)=π4 on the interval 0<x<π:
* π4=∑n=0∞cnsin⁡[nx]+dncos⁡[nx]
⇒{cn={1n(n odd)0(n even)dn=0(∀n)
5. ↑ "Bernoulli polynomials: Series representations (subsection 06/02)". http://functions.wolfram.com/Polynomials/BernoulliB2/06/02/. Retrieved 2 June 2011.
6. ↑ Hofbauer, Josef. "A simple proof of 1 + 1/22 + 1/32 + *** = π2/6 and related identities". http://homepage.univie.ac.at/josef.hofbauer/02amm.pdf. Retrieved 2 June 2011.π2/6+and+related+identities&rft.atitle=&rft.aulast=Hofbauer&rft.aufirst=Josef&rft.au=Hofbauer, Josef&rft_id=http://homepage.univie.ac.at/josef.hofbauer/02amm.pdf&rfr_id=info:sid/en.wikibooks.org:List_of_mathematical_series">
7. ↑ Sondow, Jonathan; Weisstein, Eric W.. "Riemann Zeta Function (eq. 52)". http://mathworld.wolfram.com/RiemannZetaFunction.html.
8. ↑ Abramowitz, Milton; Stegun, Irene (1964). "6.4 Polygamma functions". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Courier Corporation. p. 260. ISBN 0-486-61272-4. http://people.math.sfu.ca/~cbm/aands/page_260.htm.

## References

* Many books with a list of integrals also have a list of series.

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