Summation of Grandi's Series - Separation of Scales

Separation of Scales

Given any function φ(x) such that φ(0) = 1, the limit of φ at +∞ is 0, and the derivative of φ is integrable over (0, +∞), then the generalized φ-sum of Grandi's series exists and is equal to 1⁄2:

The Cesaro or Abel sum is recovered by letting φ be a triangular or exponential function, respectively. If φ is additionally assumed to be continuously differentiable, then the claim can be proved by applying the mean value theorem and converting the sum into an integral. Briefly:

\begin{array}{rcl} S_\varphi & = &\displaystyle \lim_{\delta\downarrow0}\sum_{m=0}^\infty\left \\ & = & \displaystyle \lim_{\delta\downarrow0}\sum_{m=0}^\infty\varphi'(2k\delta+c_k)(-\delta) \\ & = & \displaystyle-\frac12\int_0^\infty\varphi'(x) \,dx = -\frac12\varphi(x)|_0^\infty = \frac12. \end{array}

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