List of mathematical series

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
Evaluating sums
In mathematics, a series is the sum of the elements of a sequence. This article mentions a few common series and how to compute their values...

.
  • Here, is taken to have the value 1.
  • is a Bernoulli polynomial.
  • is a Bernoulli number
    Bernoulli number
    In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....

    , and here,
  • is an Euler number.
  • is the Riemann zeta function.
  • is the Gamma function
    Gamma function
    In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

    .
  • is a polygamma function.
  • is a polylogarithm.

Sums of powers


See Faulhaber's formula.

The first few values are:




See zeta constants.

The first few values are:
  • (the Basel problem
    Basel problem
    The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...

    )



Low-order polylogarithms


Finite sums:



Infinite sums, valid for (see polylogarithm):

The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form
Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...

:







Exponential function


  • (c.f. mean of Poisson distribution
    Poisson distribution
    In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...

    )

  • (c.f. second moment of Poisson distribution)




Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions


















Modified-factorial denominators




Binomial coefficients

  • (see Binomial theorem
    Binomial theorem
    In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

    )




Harmonic Numbers





Binomial coefficients


  • (see Multiset)

  • (see Vandermonde identity)

Trigonometric functions


Sums of sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

s and cosines arise in Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

.










Unclassified



See also

  • Series (mathematics)
    Series (mathematics)
    A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

  • List of integrals
  • Summation
  • Taylor series
    Taylor series
    In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

  • Binomial theorem
    Binomial theorem
    In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

  • Gregory's series
    Gregory's series
    Gregory's series, also known as the Madhava-Gregory series or Leibniz's series, is a mathematical series that was discovered by the Indian mathematician Madhava of Sangamagrama...