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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
See Faulhaber's formula.
The first few values are:
See
zeta constants
.
The first few values are:
- (the Basel problem)
Power series
Low-order polylogarithms
Finite sums:
- , (geometric series)
Infinite sums, valid for (see polylogarithm):
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
- (versine)
haversine
)
Modified-factorial denominators
- [2]
- [2]
Binomial coefficients
- (see Binomial theorem § Newton's generalized binomial theorem)
- [3]
- [3] ,
Catalan numbers
- [3] , generating function of the Central binomial coefficients
- [3]
Harmonic numbers
(See harmonic numbers, themselves defined , and generalized to the real numbers)
- [2]
- [2]
Binomial coefficients
- (see Multiset)
- (see
Vandermonde identity
)
Trigonometric functions
Sums of
.
- ,[4]
- [5]
- [6]
Rational functions
Exponential function
- (see the Landsberg–Schaar relation)
Numeric series
These numeric series can be found by plugging in numbers from the series listed above.
Alternating harmonic series
Sum of reciprocal of factorials
Trigonometry and π
Reciprocal of tetrahedral numbers
Where
Exponential and logarithms
- , that is
See also
Notes
References
- Many books with a
list of integrals
also have a list of series.