An Alternating Reciprocal Sum

★ 7/10Recursion algebra sequences Telescoping

Problem

A positive sequence $\{a_n\}$ satisfies $a_1 = \dfrac{3}{2}$ and

a_{n+1}^2 - a_n^2 = \frac{1}{(n+2)^2} - \frac{1}{n^2}.

Let $S_n$ be the sum of the first $n$ terms. Find

\frac{1}{S_1} - \frac{1}{S_3} + \frac{1}{S_5} - \cdots - \frac{1}{S_{2007}} + \frac{1}{S_{2009}}.

Answer

Solution

Difficulty	7/10
Topics	Recursion, algebra, sequences, Telescoping

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