List of mathematical series

Sums of powers

See Faulhaber's formula.

∑ k = 0 m k n − 1 = B n ( m + 1 ) − B n n

The first few values are:

∑ k = 1 m k = m ( m + 1 ) 2

∑ k = 1 m k 2 = m ( m + 1 ) ( 2 m + 1 ) 6 = m 3 3 + m 2 2 + m 6

∑ k = 1 m k 3 = [ m ( m + 1 ) 2 ] 2 = m 4 4 + m 3 2 + m 2 4

See zeta constants.

ζ ( 2 n ) = ∑ k = 1 ∞ 1 k 2 n = ( − 1 ) n + 1 B 2 n ( 2 π ) 2 n 2 ( 2 n ) !

The first few values are:

ζ ( 2 ) = ∑ k = 1 ∞ 1 k 2 = π 2 6 (the Basel problem)

ζ ( 4 ) = ∑ k = 1 ∞ 1 k 4 = π 4 90

ζ ( 6 ) = ∑ k = 1 ∞ 1 k 6 = π 6 945

Low-order polylogarithms

Finite sums:

∑ k = 0 n z k = 1 − z n + 1 1 − z , (geometric series)

∑ k = 1 n k z k = z 1 − ( n + 1 ) z n + n z n + 1 ( 1 − z ) 2

∑ k = 1 n k 2 z k = z 1 + z − ( n + 1 ) 2 z n + ( 2 n 2 + 2 n − 1 ) z n + 1 − n 2 z n + 2 ( 1 − z ) 3

∑ k = 1 n k m z k = ( z d d z ) m 1 − z n + 1 1 − z

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

Li n ⁡ ( z ) = ∑ k = 1 ∞ z k k n

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

d d z Li n ⁡ ( z ) = Li n − 1 ⁡ ( z ) z

Li 1 ⁡ ( z ) = ∑ k = 1 ∞ z k k = − ln ⁡ ( 1 − z )

Li 0 ⁡ ( z ) = ∑ k = 1 ∞ z k = z 1 − z

Li − 1 ⁡ ( z ) = ∑ k = 1 ∞ k z k = z ( 1 − z ) 2

Li − 2 ⁡ ( z ) = ∑ k = 1 ∞ k 2 z k = z ( 1 + z ) ( 1 − z ) 3

Li − 3 ⁡ ( z ) = ∑ k = 1 ∞ k 3 z k = z ( 1 + 4 z + z 2 ) ( 1 − z ) 4

Li − 4 ⁡ ( z ) = ∑ k = 1 ∞ k 4 z k = z ( 1 + z ) ( 1 + 10 z + z 2 ) ( 1 − z ) 5

Exponential function

∑ k = 0 ∞ z k k ! = e z

∑ k = 0 ∞ k z k k ! = z e z (cf. mean of Poisson distribution)

∑ k = 0 ∞ k 2 z k k ! = ( z + z 2 ) e z (cf. second moment of Poisson distribution)

∑ k = 0 ∞ k 3 z k k ! = ( z + 3 z 2 + z 3 ) e z

∑ k = 0 ∞ k 4 z k k ! = ( z + 7 z 2 + 6 z 3 + z 4 ) e z

∑ k = 0 ∞ k n z k k ! = z d d z ∑ k = 0 ∞ k n − 1 z k k ! = e z T n ( z )

where T n ( z ) is the Touchard polynomials.

Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions

∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) ! = sin ⁡ z

∑ k = 0 ∞ z 2 k + 1 ( 2 k + 1 ) ! = sinh ⁡ z

∑ k = 0 ∞ ( − 1 ) k z 2 k ( 2 k ) ! = cos ⁡ z

∑ k = 0 ∞ z 2 k ( 2 k ) ! = cosh ⁡ z

∑ k = 1 ∞ ( − 1 ) k − 1 ( 2 2 k − 1 ) 2 2 k B 2 k z 2 k − 1 ( 2 k ) ! = tan ⁡ z , | z | < π 2

∑ k = 1 ∞ ( 2 2 k − 1 ) 2 2 k B 2 k z 2 k − 1 ( 2 k ) ! = tanh ⁡ z , | z | < π 2

∑ k = 0 ∞ ( − 1 ) k 2 2 k B 2 k z 2 k − 1 ( 2 k ) ! = cot ⁡ z , | z | < π

∑ k = 0 ∞ 2 2 k B 2 k z 2 k − 1 ( 2 k ) ! = coth ⁡ z , | z | < π

∑ k = 0 ∞ ( − 1 ) k − 1 ( 2 2 k − 2 ) B 2 k z 2 k − 1 ( 2 k ) ! = csc ⁡ z , | z | < π

∑ k = 0 ∞ − ( 2 2 k − 2 ) B 2 k z 2 k − 1 ( 2 k ) ! = csch ⁡ z , | z | < π

∑ k = 0 ∞ ( − 1 ) k E 2 k z 2 k ( 2 k ) ! = sec ⁡ z , | z | < π 2

∑ k = 0 ∞ E 2 k z 2 k ( 2 k ) ! = sech ⁡ z , | z | < π 2

∑ k = 1 ∞ ( − 1 ) k − 1 z 2 k ( 2 k ) ! = ver ⁡ z (versine)

∑ k = 1 ∞ ( − 1 ) k − 1 z 2 k 2 ( 2 k ) ! = hav ⁡ z (haversine)

∑ k = 0 ∞ ( 2 k ) ! z 2 k + 1 2 2 k ( k ! ) 2 ( 2 k + 1 ) = arcsin ⁡ z , | z | ≤ 1

∑ k = 0 ∞ ( − 1 ) k ( 2 k ) ! z 2 k + 1 2 2 k ( k ! ) 2 ( 2 k + 1 ) = arcsinh ⁡ z , | z | ≤ 1

∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 2 k + 1 = arctan ⁡ z , | z | < 1

∑ k = 0 ∞ z 2 k + 1 2 k + 1 = arctanh ⁡ z , | z | < 1

ln ⁡ 2 + ∑ k = 1 ∞ ( − 1 ) k − 1 ( 2 k ) ! z 2 k 2 2 k + 1 k ( k ! ) 2 = ln ⁡ ( 1 + 1 + z 2 ) , | z | ≤ 1

Modified-factorial denominators

∑ k = 0 ∞ ( 4 k ) ! 2 4 k 2 ( 2 k ) ! ( 2 k + 1 ) ! z k = 1 − 1 − z z , | z | < 1

∑ k = 0 ∞ 2 2 k ( k ! ) 2 ( k + 1 ) ( 2 k + 1 ) ! z 2 k + 2 = ( arcsin ⁡ z ) 2 , | z | ≤ 1

∑ n = 0 ∞ ∏ k = 0 n − 1 ( 4 k 2 + α 2 ) ( 2 n ) ! z 2 n + ∑ n = 0 ∞ α ∏ k = 0 n − 1 [ ( 2 k + 1 ) 2 + α 2 ] ( 2 n + 1 ) ! z 2 n + 1 = e α arcsin ⁡ z , | z | ≤ 1

Binomial coefficients

( 1 + z ) α = ∑ k = 0 ∞ ( α k ) z k , | z | < 1 (see Binomial theorem)

∑ k = 0 ∞ ( α + k − 1 k ) z k = 1 ( 1 − z ) α , | z | < 1

∑ k = 0 ∞ 1 k + 1 ( 2 k k ) z k = 1 − 1 − 4 z 2 z , | z | ≤ 1 4 , generating function of the Catalan numbers

∑ k = 0 ∞ ( 2 k k ) z k = 1 1 − 4 z , | z | < 1 4 , generating function of the Central binomial coefficients

∑ k = 0 ∞ ( 2 k + α k ) z k = 1 1 − 4 z ( 1 − 1 − 4 z 2 z ) α , | z | < 1 4

Harmonic numbers

∑ k = 1 ∞ H k z k = − ln ⁡ ( 1 − z ) 1 − z , | z | < 1

∑ k = 1 ∞ H k k + 1 z k + 1 = 1 2 [ ln ⁡ ( 1 − z ) ] 2 , | z | < 1

∑ k = 1 ∞ ( − 1 ) k − 1 H 2 k 2 k + 1 z 2 k + 1 = 1 2 arctan ⁡ z log ⁡ ( 1 + z 2 ) , | z | < 1

∑ n = 0 ∞ ∑ k = 0 2 n ( − 1 ) k 2 k + 1 z 4 n + 2 4 n + 2 = 1 4 arctan ⁡ z log ⁡ 1 + z 1 − z , | z | < 1

Binomial coefficients

∑ k = 0 n ( n k ) = 2 n

∑ k = 0 n ( − 1 ) k ( n k ) = 0 ,  where  n > 0

∑ k = 0 n ( k m ) = ( n + 1 m + 1 )

∑ k = 0 n ( m + k − 1 k ) = ( n + m n ) (see Multiset)

∑ k = 0 n ( α k ) ( β n − k ) = ( α + β n ) (see Vandermonde identity)

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

∑ k = 1 ∞ sin ⁡ ( k θ ) k = π − θ 2 , 0 < θ < 2 π

∑ k = 1 ∞ cos ⁡ ( k θ ) k = − 1 2 ln ⁡ ( 2 − 2 cos ⁡ θ ) , θ ∈ R

∑ k = 0 ∞ sin ⁡ [ ( 2 k + 1 ) θ ] 2 k + 1 = π 4 , 0 < θ < π

B n ( x ) = − n ! 2 n − 1 π n ∑ k = 1 ∞ 1 k n cos ⁡ ( 2 π k x − π n 2 ) , 0 < x < 1

∑ k = 0 n sin ⁡ ( θ + k α ) = sin ⁡ ( n + 1 ) α 2 sin ⁡ ( θ + n α 2 ) sin ⁡ α 2

∑ k = 1 n − 1 sin ⁡ π k n = cot ⁡ π 2 n

∑ k = 1 n − 1 sin ⁡ 2 π k n = 0

∑ k = 0 n − 1 csc 2 ⁡ ( θ + π k n ) = n 2 csc 2 ⁡ ( n θ )

∑ k = 1 n − 1 csc 2 ⁡ π k n = n 2 − 1 3

∑ k = 1 n − 1 csc 4 ⁡ π k n = n 4 + 10 n 2 − 11 45

Rational functions

∑ m = b + 1 ∞ b m 2 − b 2 = 1 2 H 2 b

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. 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.