We derive a new technique for evaluating or approximating sums by means of complex integration. Our result is sufficiently general that it is applicable to a wide variety of functions. We consider examples that illustrate the power of the technique: our first example is an alternative derivation of the Euler–Maclaurin sum formula for the case in which the remainder term vanishes, and our other two examples show how our technique can be applied when the Euler–Maclaurin formula is not useful.

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