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.
See Faulhaber's formula.
The first few values are:
See zeta constants.
The first few values are:

  • Power series

Low-order polylogarithms

Finite sums:
Infinite sums, valid for :
The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form:

Exponential function

where is the Touchard polynomials.

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

  • Modified-factorial denominators

Binomial coefficients

Binomial coefficients

Sums of sines and cosines arise in Fourier series.
  • ,

    Roots of unity

A 'th root of unity is a solution to the equation and they can be written like:
The following summation identities hold:
Let be an integer then we also got:

Rational functions

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

Alternating harmonic series

Alternating arithmetic series

Let be defined as:
where are positive whole numbers. Then if we can write and, where, and get:
Now if we can, per Euclid's division lemma, write where and then
where we now can add the remaining rows back and subtract them to give us:
what that means is that all the infinite choices of and can essentially be boiled down to the cases where and. If we assume those two things we can then write:
and in the case of using a negative sign instead:
the same two rules apply from above apply and then we can do the following for the case with :
Let us test out the formula:

Sum of reciprocal of factorials

Trigonometry and π

Reciprocal of tetrahedral numbers

Where

Exponential and logarithms

  • , that is