# Prove the Famous Result: $$\sum_{k = 1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$$

• MHB
• Euge
In summary, the "Famous Result" is the value of the infinite series given by the equation $$\sum_{k = 1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$$. This value is calculated using the Euler-Maclaurin summation formula and can also be proven using other methods such as Fourier series, complex analysis, and the Riemann zeta function. It has important applications in number theory, complex analysis, and the theory of partitions, and is considered "famous" due to its historical significance and implications in mathematics.
Euge
Gold Member
MHB
POTW Director
Hi MHB Community,

I'm sorry I haven't been around. For several months I've been very sick. I wish you all a Happy New Year! In respect of the MHB equations above, here's a good problem to start the new year:

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Prove the famous result $\sum_{k = 1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$ Use any method(s) you like.

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No one answered this week's problem. You can read my solution below.
Apply the Parseval identity to the function $f(x) = x$, $-\pi \le x \le \pi$:
$\sum_{-\infty}^\infty \lvert c_n\rvert^2 = \frac{1}{2\pi} \int_{-\pi}^\pi \lvert f(x)\rvert^2\, dx$
where the $c_n$ are the coefficients of the complex Fourier series of $f$. We have $$c_n = \frac{1}{2\pi}\int_{-\pi}^\pi x e^{-inx}\, dx$$, so that $c_0 = 0$ and for $n\neq 0$, integration by parts yields $c_n = \frac{1}{2\pi}\left\{x\left(-\frac{1}{in}e^{-inx}\right) - \left(-\frac{1}{n^2}e^{-inx}\right)\right\}\Bigl|_{x = -L}^L = \frac{(-1)^{n+1}}{in}$ The Parseval equation above reduces to $2\sum_{n = 1}^\infty \frac{1}{n^2} = \frac{1}{2\pi}\int_{-\pi}^\pi x^2\, dx$ or $2\sum_{n = 1}^\infty \frac{1}{n^2} = \frac{\pi^2}{3}$ Therefore, $\sum_{n = 1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$

## What is the famous result $$\sum_{k = 1}^\infty \frac{1}{k^2}$$?

The famous result $$\sum_{k = 1}^\infty \frac{1}{k^2}$$ is known as the Basel problem, named after the Swiss mathematician Leonhard Euler who first proved it in 1735. It is a special case of the more general result known as the Riemann zeta function, and it states that the sum of the reciprocals of the squares of all positive integers converges to $$\frac{\pi^2}{6}$$.

## Why is this result considered famous?

This result is considered famous because it is a surprising and elegant solution to a seemingly simple problem. It also has connections to many other areas of mathematics, including number theory, complex analysis, and probability theory.

The result was first proved by Euler using a technique known as analytic continuation. Essentially, he showed that the sum of the reciprocals of the squares can be expressed as a complex function, and then used properties of this function to evaluate the sum. Later, other mathematicians such as Bernhard Riemann and Carl Friedrich Gauss provided alternative proofs using different methods.

## Can this result be generalized to other values?

Yes, this result can be generalized to other values of the exponent. For example, the sum of the reciprocals of the cubes of all positive integers converges to $$\zeta(3) = \frac{\pi^3}{180}$$, and in general, the sum of the reciprocals of the $$n$$th powers of all positive integers converges to $$\zeta(n)$$. However, the exact values of these sums for other values of $$n$$ are not known, except for a few special cases.

## What practical applications does this result have?

While this result may not have any direct practical applications, it has been used in various fields of mathematics and physics. For example, it has been used to prove the irrationality of certain numbers and to study the behavior of certain systems with infinitely many components. It has also been used in statistical mechanics to calculate the average energy of a quantum harmonic oscillator.

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