Anonymous asked in Science & MathematicsMathematics · 1 decade ago

How are infinite sums evaluated?


 ∑ 1/i²= 1/1² + 1/2² + 1/3² + …


This somehow evaluates to: π²/6

1 Answer

  • Duke
    Lv 7
    1 decade ago
    Favorite 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.

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