A neat enigma

Calculus Level 3

∑ n = 1 ∞ 1 a n 2 = ∫ 0 1 d x 4 − x 2 \displaystyle \sum _{ n=1 }^{ \infty }{ \frac { 1 }{ a{ n }^{ 2 } } =\int _{ 0 }^{ 1 }{ \frac { dx }{ \sqrt { 4-{ x }^{ 2 } } } } }

Find the value of a a .

This problem is original.

Picture credits: Rainbow 02 by Jerry Magnum Porsbjer, Wikipedia

3 π 2 \frac{3}{\pi ^2} 1 1 3 π 3 \pi 1 π 2 \frac{1}{\pi ^2} π \pi No solution

Jonas Katona by Jonas Katona , 22, USA

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1 solution

Tom Capizzi
Aug 20, 2016

Euler found the sum of the reciprocal of squares to be Pi^2/6, so LHS = 1/a * Pi^2/6. The definite integral is a standard form, and it is arcsin (x/2) evaluated between the limits of 1 and 0. (The substitution x = 2 cos (theta) simplifies the expression to -Integral d(theta) from Pi/3 to Pi/2.) RHS = Pi/6 = 1/a * Pi^2/6. Therefore, a = Pi.

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