= Meromorphic telescoping cancellation
{title2=$\sum_{k=1}^N[f(w+k\delta)-f(w+(k-1)\delta)]=f(w+N\delta)-f(w)$}
For a <meromorphic function> $f$, the displayed <telescoping sum> identity holds wherever every summand is defined. If the two endpoint terms are finite at $w=w_0$, their difference gives a <holomorphic> continuation of the entire sum to $w_0$, even when individual summands have <poles> there. This continued value is a common-shift <limit>; it does not make an originally undefined sum of <complex numbers> well defined without an additional convention.
The proof is finite cancellation on the common domain of definition, followed by <analytic continuation>. For $f(w)=\tanh w$, a common shift away from the imaginary axis avoids its <poles>. Different independently chosen shifts of individual summands need not give the same <limit>, so specifying the common shift matters.
Back to article page
