The value of the series $\sum_{n=1}^\infty \frac{1}{n^2} $ is well-known to be $\pi^2/6$ and there are many proofs of this http://en.wikipedia.org/wiki/Basel_problem. How can one show that the seemingly related series $\sum_{n=3}^\infty \frac{1}{n^2 - 4}$ has sum $25/48$?
Exact value of a series question
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$\begingroup$
sequences-and-series
analysis
1 Answers
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Hint: $\dfrac{1}{n-2}-\dfrac{1}{n+2} = \dfrac{4}{n^2-4}$
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0Thank you Thomas. Only the first four $p$ositive terms survive and the sum is 1/4(1 + 1/2 + 1/3 + 1/4) = 25/48 – 2012-04-19