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 of, separation and/or scales:

    “... the separation of church and state means separation—absolute and eternal—or it means nothing.”
    —Agnes E. Meyer (1887–1970)

    “Like sleep disturbances, some worries at separation can be expected in the second year. If you accept this, then you will avoid reacting to this anxiety as if it’s your fault. A mother who feels guilty will appear anxious to the child, as if to affirm the child’s anxiety. By contrast, a parent who understands that separation anxiety is normal is more likely to react in a way that soothes and reassures the child.”
    —Cathy Rindner Tempelsman (20th century)

    “As deaths have accumulated I have begun to think of life and death as a set of balance scales. When one is young, the scale is heavily tipped toward the living. With the first death, the first consciousness of death, the counter scale begins to fall. Death by death, the scales shift weight until what was unthinkable becomes merely a matter of gravity and the fall into death becomes an easy step.”
    —Alison Hawthorne Deming (b. 1946)