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}

Read more about this topic:  Summation Of Grandi's Series

Famous quotes containing the words separation and/or scales:

    I was the one who was working to destroy the one thing to which I was committed, that is, my relationship with Gilberte; I was doing so by creating, little by little and through the prolonged separation from my friend, not her indifference, but my own. It was toward a long and cruel suicide of the self within myself which loved Gilberte that I continuously set myself ...
    Marcel Proust (1871–1922)

    It cannot but affect our philosophy favorably to be reminded of these shoals of migratory fishes, of salmon, shad, alewives, marsh-bankers, and others, which penetrate up the innumerable rivers of our coast in the spring, even to the interior lakes, their scales gleaming in the sun; and again, of the fry which in still greater numbers wend their way downward to the sea.
    Henry David Thoreau (1817–1862)