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Prove that the sum from 1 to ∞ of ((n+1)1/2 – n1/2) / (n2 + n)1/2 = 1

by RoRi

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

    \[ \sum_{n=1}^{\infty} \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n^2+n}} = 1. \]

We simplify the terms and then apply the theorem on sums of telescoping series (Theorem 10.7, page 386 of Apostol),

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

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