The Famous Basel Problem: The process that led Euler to the answer!

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Euler is considered to be one of the greatest mathematician of the 18th century and many believe that he was among the greatest of all time.

The answer to the Basel problem is considered to be among his many gems. It is the sum of the series shown below and the answer is astounding!
What is π doing there?

Mathematician at the time summed up the series to the first 10, 100, and 1000 terms and the slow rate of convergence gave very little clue to what the answer could be. 

 Euler realized that he would have to find a function that would converge much faster. After some trial and error he found the function shown below whose power series expansion is shown.
 So if we integrate the above expression from 0 to 1


This is what he was looking for!
To evaluate the integral in the left, he broke it into two parts.
Then he evaluated each of the integrals in turn.

Then the next integral


Put x=1-t
When x=1/2, t=1/2
When x=1, t=0
dx = -dt
We know

 This is an improper integral which we can evaluate in a calculator.
Simplifying and using 

Now when we find the sum of the first 10, 100 and 1000 terms it converges very quickly and the first 11 digits are the same.


This gave the required conviction to Euler that the solution was π2/6 and he will be able to prove it!

His famous proof is documented at several places and here are the key steps.

Comparing the x3 term 

Hope you enjoyed the story behind the Basel Problem.

 Reference
1) Euler:The Master of Us All, William Dunham (1999)

2) The Mathematics of Euler, Simon Rubinstein-Salzedo, Euler Circle.

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