I have a doubt regarding the Abel Plana summation formula.
What is the condition to apply the Abel Plana summation formula.??
This question arises because sometimes while evaluating series it gives exact results example given
$$\sum_{k=1}^\infty \frac{\sin k}{k}=\frac{\pi -1}{2}$$
However applying the same formula doesn't gives the correct result for some functions example given :
$$\sum_{k=1}^{\infty} e^{-k^2} \neq \frac{\sqrt{\pi }-1}{2} $$
But this sum indeed is given in terms of Jacobi theta function. However the closed form on right still gives a seemingly nice approximation for the sum.
I guess I'm missing something very silly but any information in simple terms for what function f does the formula can be applied.
Thank you !)
2026-03-26 16:11:30.1774541490
Doubt regarding Abel Plana summation formula
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