I ran across this behaviour that I do not understand. I wanted to prove a formula for a known convergent series:
$\sum_{k=-\infty}^\infty \frac{1}{(2k+1)(2q-2k-1)} = -\frac{\pi^2}{4} \delta_{q,0}\quad$ for $\quad q\in\mathbb{Z}$.
However, Mathematica says this is wrong:
Resolve[ForAll[q, Sum[1/((2 k + 1) (2 q - 2 k - 1)) == -(\[Pi]^2/4) KroneckerDelta[q, 0], {k,-Infinity,Infinity}], Integers]
returns False
. Moreover, if the values for $q$ are restricted, i.e. if I look separately at the cases $q=0$ and $q\neq 0$, I get a True
result: The two commands
Resolve[ForAll[q, q == 0, Sum[1/((2 k + 1) (2 q - 2 k - 1)) == -(\[Pi]^2/4) KroneckerDelta[q, 0], {k,-Infinity,Infinity}], Integers]
Resolve[ForAll[q, q != 0, Sum[1/((2 k + 1) (2 q - 2 k - 1)) == -(\[Pi]^2/4) KroneckerDelta[q, 0], {k,-Infinity,Infinity}], Integers]
both evaluate to True
. Logically, this is a contradiction, so my question is whether I misunderstood the function of these commands or this is a bug. Thanks in advance for clarification! (Using Mathematica 10.0)
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