The Basel problem is an issue of Pietro Mengoli’s theory of numbers in 1644 and Leonhard Euler’s resolution in 1734. Since Euler’s solution stayed open for 90 years, at the age of 28 he was immediately renowned for discovering solutions to this problem.
This problem asks for the sum of the inverse of the square number. Initially, it may sound somewhat confusing, but after reading this article you will be clear about concept and approach to this problem.

What is the Basel Problem?

The Basel Problem is a famous question in the history of mathematics, named after the city of Basel in Switzerland, where the prominent mathematician Leonhard Euler studied. The problem asks for the exact sum of the reciprocals of the squares of the natural numbers:

[Tex]\sum_{n=1}^{\infty} \frac{1}{n^2} [/Tex]

Let’s define k first, as it will help us initially with the problem. So here’s k…

k=1

Here k is an abbreviation. Now you need to find the inverse sum of this ‘k’. This problem has five different ways of solving it.
There is a certain probability that drags this problem into the complicated section.

How to Find Inverse using Basel Problem?

Finding inverse, there are several different ways, and certainly, the Basel way is one of the best possible ways to find the inverse sum.

• Just assume that you have an abbreviation and then simply add a pi value to the base of it. 
• Then add it to the original question. 

Sounds Simple, Right? Well, indeed it is. Because the way of solving it is radically easy, some great mathematicians called it the fool’s way of solving the problem, however, the majority found it fascinating as they did not have access to such powerful scientific calculators at that time.
The release of this problem garnered a lot of attention. People never had to do the advanced calculations neither had to invest in hiring metrics, all they had to do was to provide values, inverse them out, and add to the original term! 
Euler could be called the main person behind the development of a solution to this problem, thus he got immediate attention to it.

Proof

We have, sin x = x - x³/3! + x⁵/5! - x⁷/7! + ... sin x /x = 1 - x²/3! + x⁴/5! - x⁶/7! + ... Hence, » sinx/x (1-pi/x)

As you may see, that changing certain values and transposing the sin x we can quickly have the inverse of a function and that too without much calculation, something which was not easy back in the time when there were no computers.

This in turn proved that humans could calculate complex inverse functions without the need for specific hardware, something which was amazingly fantastic at that time!

Linkage to the Riemann Zeta Function

The Riemann zeta function is one of the most significant functions in mathematics because of its relationship to distribution of prime numbers. And the solution for this Basel problem provided a conjunction point for connecting Riemann zeta function to this problem. It had major implications at that time. Eventually, mathematicians also had to admit its efficiency and they admitted it. This was not first time but, when Euler didn’t have a certain justification for a problem. Just like every other proof, he even proved existing solution to this problem!

Read More: Riemann Sums

Conclusion – Basel Problem

Euler was the first person who devised a method for actually solving this problem, and since back then it’s due to him that we have actually arrived at a possible solution to Basel Problem. The trick might sound absurd at this time because of technical advancements in science and technology, but it was a great discovery back at that time!

FAQs on Basel Problem

What is the Basel Problem?

The Basel Problem asks for the exact sum of the infinite series of the reciprocals of the squares of the natural numbers, given by [Tex]\sum_{n=1}^{\infty} \frac{1}{n^2} [/Tex].

Who solved the Basel Problem and when?

The Basel Problem was solved by Leonhard Euler in 1734, who showed that the sum of the series is π²/6.

Why is the Basel Problem important in mathematics?

The Basel Problem is important because its solution demonstrated advanced techniques in analysis and laid the groundwork for further developments in the study of series and the Riemann zeta function.

How did Euler solve the Basel Problem?

Euler used the Taylor series expansion of the sine function and innovative factorization methods to equate coefficients in infinite series expansions, leading to the solution π²/6.