Summation Formula for Alternating Series

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Theorem

Let $C_N$ be the square with vertices $\paren {N + \dfrac 1 2} \paren {\pm 1 \pm i}$ for some real $N \in \N$.

Let $f$ be a function meromorphic on $C_N$.

Let $\cmod {\map f z} < \dfrac M {\cmod z^k}$, for constants $k > 1$ and $M$ independent of $N$, for all $z \in \partial C_N$.

Let $X$ be the set of poles of $f$.

Then:

$\displaystyle \sum_{n \mathop \in \Z \mathop \setminus X} \paren {-1}^n \map f n = -\sum_{z_0 \mathop \in X} \Res {\pi \map \csc {\pi z} \map f z} {z_0}$

If $X \cap \Z = \O$, this becomes:

$\displaystyle \sum_{n \mathop = -\infty}^\infty \paren {-1}^n \map f n = -\sum_{z_0 \mathop \in X} \Res {\pi \map \csc {\pi z} \map f z} {z_0}$


Proof


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