Basel Problem: Pi Squared Over 6 and Infinite Series
Published on August 23, 2024
Question
how to proof it
Answer
This is the famous Basel Problem, first posed in the 17th century and solved by Leonhard Euler.
It’s not something you can prove with basic algebra or calculus. It requires more advanced mathematical tools like Fourier series or complex analysis.
Here’s a brief outline of how the proof using Fourier series works:
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Start with the Fourier series: The Fourier series allows you to represent a periodic function as an infinite sum of sines and cosines.
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Choose a specific function: Consider the function f(x) = x^2 on the interval [-π, π], and extend it periodically to the entire real line.
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Find its Fourier series: Calculate the Fourier coefficients for this function.
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Apply Parseval’s Identity: Parseval’s identity relates the integral of a function squared to the sum of the squares of its Fourier coefficients.
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Evaluate at a special point: Evaluate both sides of Parseval’s identity at x = 0. This will lead to a series involving the reciprocals of squares.
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Manipulate the series: Through some clever manipulations, you can transform the series into the desired form (1/1² + 1/2² + 1/3² + …).
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The result: The final result will be the equation you asked for: π²/6 = 1/1² + 1/2² + 1/3² + …
Let me know if you’d like to see a more detailed explanation of a specific step, but be warned that a full proof can be quite involved.