 Meromorphic telescoping cancellation

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.