Odd Power

Author

Eric W. Weisstein
November 30, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/OddPower.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/OddPower.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Sum

One can sum the reciprocals of the odd powers of the integers that are 3 (mod 4).

(∑_(n=0)^∞)(∑_(m=1)^∞)(1/(((4 n+3))^(2 m+1)))==(1/8) (π-4 log(2))

V6

Sum[1/(4n+3)^(2m+1),{n,0,∞},{m,∞}]

V7

Sum[1/(4n+3)^(2m+1),{n,0,∞},{m,∞}]

FullSimplify[%]

V8

$Version

Sum[1/(4n+3)^(2m+1),{n,0,∞},{m,∞}]

(∑_(n=0)^∞)(1/(((4 n+3))^(2 m+1)))==(4^(-2 m-1)) ζ(2 m+1,(3/4))

V6

Sum[1/(4n+3)^(2m+1),{n,0,∞}]

Sum[%,{m,∞}]

V7

Sum[1/(4n+3)^(2m+1),{n,0,∞}]

Sum[%,{m,∞}]

(∑_(m=1)^∞)(1/(((4 n+3))^(2 m+1)))==(1/(8 (4 n+3) (2 (n^2)+3 n+1)))

V6

Sum[1/(4n+3)^(2m+1),{m,∞}]

Sum[%,{n,0,∞}]

V6

Sum[1/(4n+3)^(2m+1),{m,∞}]

Sum[%,{n,0,∞}]

FullSimplify[%]