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Prove that the sum from 1 to ∞ of (2n + 1) / (n2(n+1)2) = 1

by RoRi

Prove that the following sum converges and has the given value.

    \[ \sum_{n=1}^{\infty} \frac{2n+1}{n^2 (n+1)^2} = 1. \]

Again, we simplify and use the theorem on the sum of a telescoping series (Theorem 10.7 on page 386 of Apostol),

    \begin{align*} \sum_{n=1}^{\infty} \frac{2n+1}{n^2(n+1)^2} &= \sum_{n=1}^{\infty} \frac{n^2 + 2n + 1 - n^2}{n^2(n+1)^2} \\[9pt] &= \sum_{n=1}^{\infty} \left( \frac{1}{n^2} - \frac{1}{(n+1)^2} \right) \\[9pt] &= \sum_{n=1}^{\infty} (b_n - b_{n+1}) &(b_n = \frac{1}{n^2}) \\[9pt] &= b_1 - L \\[9pt] &= 1. \end{align*}

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