The Basel Problem

The Basel Problem is a famous problem in mathematics that was first posed by Pietro Mengoli in 1644 and was later solved by Leonhard Euler in 1735. The problem asks to find the exact value of the infinite sum of the reciprocal squares of natural numbers. In other words, we are asked to find the value of the following series:

$\sum_{n=1}^\infty \frac{1}{n^2}$

The problem has applications in various fields of mathematics, including number theory, analysis, and probability theory.

Euler's Solution

Euler was able to solve the Basel problem by using a clever technique called the Euler-Maclaurin formula, which allows us to approximate a sum as an integral. Here's a brief overview of Euler's solution:

First, we start by writing down the Taylor series expansion of the function $f(x) = \frac{1}{x^2}$ around the point $x = 1$:

$f(x) = \frac{1}{x^2} = \sum_{n=0}^\infty (-1)^n\frac{(n+1)x^n}{n!}$

Next, we integrate this series term-by-term from $x = 1$ to $x = \infty$:

$\begin{aligned} \int_1^\infty \frac{1}{x^2} dx &= \int_1^\infty \sum_{n=0}^\infty (-1)^n\frac{(n+1)x^n}{n!} dx \\ &= \sum_{n=0}^\infty (-1)^n\frac{(n+1)}{n!} \int_1^\infty x^n dx \\ &= \sum_{n=0}^\infty (-1)^n\frac{(n+1)}{n!} \cdot \frac{1}{n+1} \\ &= \sum_{n=0}^\infty (-1)^n\frac{1}{n!} \\ &= e^{-1} \end{aligned}$

Finally, we use the fact that the integral we just computed is equal to the sum we are interested in:

$\begin{aligned} \sum_{n=1}^\infty \frac{1}{n^2} &= \int_1^\infty \frac{1}{x^2} dx \\ &= e^{-1} \\ &= \frac{1}{e} \end{aligned}$

Therefore, the exact value of the infinite sum of the reciprocal squares of natural numbers is $\frac{1}{e}$.

Conclusion

The Basel problem is a classic example of how a simple-looking problem can sometimes have a surprisingly difficult solution. Euler's solution to the problem was not only elegant but also helped pave the way for the development of many important techniques in mathematics.

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バーゼル問題[JA]