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, 0 0 is taken to have the value 1 B n ( x ) is a Bernoulli polynomial. B n is a Bernoulli number, and here, B 1 = − 1 2 . E n is an Euler number. ζ ( s ) is the Riemann zeta function. Γ ( z ) is the gamma function. ψ n ( z ) is a polygamma function. Li s ( z ) is a polylogarithm.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 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 ∑ 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 ∑ 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 ( 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 ∑ 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 ∑ 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)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 ∑ 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.