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Question: simplification of a sum of a series

Question:simplification of a sum of a series

Lonely 400 Maple
infinity
April 01 2009
0

we want to find the sum:

sum((6*n+5)*(6*n+3)/((12*n*n+12*n+1)*(6*n*n+6*n+1)), n = 1 .. infinity)

is is equal to:

-15-(1/2)*sqrt(3)*Pi*tan((1/6)*Pi*sqrt(3))+(1/2)*sqrt(6)*Pi*tan((1/6)*Pi*sqrt(6))-2*Psi(1/2+(1/6)*sqrt(6))+Pi*tan((1/6)*Pi*sqrt(6))+2*Psi(1/2+(1/6)*sqrt(3))-Pi*tan((1/6)*Pi*sqrt(3))

can we get rid of  the function "\Psi"?  Can we bound this function \Psi with other known constant such as zeta function?

 
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