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# How are infinite sums evaluated?

∞

∑ 1/i²= 1/1² + 1/2² + 1/3² + …

i=1

This somehow evaluates to: π²/6

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- DukeLv 71 decade agoFavorite Answer
There are too many methods, the subject can hardly be covered in an answer like this. As far as the series in Your question /this is one of the famous Euler's results/ is concerned, maybe the shortest way I know is the following: express the function x² /-π ≤ x ≤ π/ as a Fourier series, calculating its coefficients by the Euler-Fourier formulas /see the link below; x², being an even function will have only cosines with non-zero coefficients/:

x² = π²/3 + 4 ∑[k=1 to ∞] (-1)^k * cos(kx) / k²

Now take x = π and You obtain the result You need.

Source(s): http://en.wikipedia.org/wiki/Fourier_series

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