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1: 4.11 Sums
For infinite series involving logarithms and/or exponentials, see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §44), and Prudnikov et al. (1986a, Chapter 5).
2: 25.8 Sums
25.8.1 k = 2 ( ζ ( k ) - 1 ) = 1 .
25.8.2 k = 0 Γ ( s + k ) ( k + 1 ) ! ( ζ ( s + k ) - 1 ) = Γ ( s - 1 ) , s 1 , 0 , - 1 , - 2 , .
25.8.5 k = 2 ζ ( k ) z k = - γ z - z ψ ( 1 - z ) , | z | < 1 .
25.8.9 k = 1 ζ ( 2 k ) ( 2 k + 1 ) 2 2 k = 1 2 - 1 2 ln 2 .
25.8.10 k = 1 ζ ( 2 k ) ( 2 k + 1 ) ( 2 k + 2 ) 2 2 k = 1 4 - 7 4 π 2 ζ ( 3 ) .
3: 8.15 Sums
For sums of infinite series whose terms include incomplete gamma functions, see Prudnikov et al. (1986b, §5.2).
4: 24.8 Series Expansions
§24.8(i) Fourier Series
If n = 1 , 2 , and 0 x 1 , then …
§24.8(ii) Other Series
24.8.9 E 2 n = ( - 1 ) n k = 1 k 2 n cosh ( 1 2 π k ) - 4 k = 0 ( - 1 ) k ( 2 k + 1 ) 2 n e 2 π ( 2 k + 1 ) - 1 , n = 1 , 2 , .
5: 1.9 Calculus of a Complex Variable
§1.9(v) Infinite Sequences and Series
Weierstrass M -test
§1.9(vii) Inversion of Limits
Dominated Convergence Theorem
6: 15.15 Sums
For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975). …
7: 25.16 Mathematical Applications
25.16.1 ψ ( x ) = m = 1 p m x ln p ,
25.16.5 H ( s ) = n = 1 h ( n ) n s ,
25.16.13 n = 1 ( h ( n ) n ) 2 = 17 4 ζ ( 4 ) ,
25.16.14 r = 1 k = 1 r 1 r k ( r + k ) = 5 4 ζ ( 3 ) ,
25.16.15 r = 1 k = 1 r 1 r 2 ( r + k ) = 3 4 ζ ( 3 ) .
8: 16.11 Asymptotic Expansions
For subsequent use we define two formal infinite series, E p , q ( z ) and H p , q ( z ) , as follows:
16.11.1 E p , q ( z ) = ( 2 π ) ( p - q ) / 2 κ - ν - ( 1 / 2 ) e κ z 1 / κ k = 0 c k ( κ z 1 / κ ) ν - k , p < q + 1 ,
16.11.2 H p , q ( z ) = m = 1 p k = 0 ( - 1 ) k k ! Γ ( a m + k ) ( = 1 m p Γ ( a - a m - k ) / = 1 q Γ ( b - a m - k ) ) z - a m - k .
9: 25.2 Definition and Expansions
25.2.1 ζ ( s ) = n = 1 1 n s .
§25.2(ii) Other Infinite Series
25.2.2 ζ ( s ) = 1 1 - 2 - s n = 0 1 ( 2 n + 1 ) s , s > 1 .
25.2.3 ζ ( s ) = 1 1 - 2 1 - s n = 1 ( - 1 ) n - 1 n s , s > 0 .
25.2.6 ζ ( s ) = - n = 2 ( ln n ) n - s , s > 1 .
10: 25.12 Polylogarithms
25.12.7 Li 2 ( e i θ ) = n = 1 cos ( n θ ) n 2 + i n = 1 sin ( n θ ) n 2 .
25.12.8 n = 1 cos ( n θ ) n 2 = π 2 6 - π θ 2 + θ 2 4 .
25.12.9 n = 1 sin ( n θ ) n 2 = - 0 θ ln ( 2 sin ( 1 2 x ) ) d x .
25.12.10 Li s ( z ) = n = 1 z n n s .
25.12.12 Li s ( z ) = Γ ( 1 - s ) ( ln 1 z ) s - 1 + n = 0 ζ ( s - n ) ( ln z ) n n ! , s 1 , 2 , 3 , , | ln z | < 2 π ,