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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 How can one show that the seemingly related series $$\sum_{n=3}^\infty \frac{1}{n^2 - 4}$$ has sum $25/48$?

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Hint: $\dfrac{1}{n-2}-\dfrac{1}{n+2} = \dfrac{4}{n^2-4}$

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Thank you Thomas. Only the first four positive terms survive and the sum is 1/4(1 + 1/2 + 1/3 + 1/4) = 25/48 – Martin Apr 19 '12 at 19:23

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