Euler’s Series: problem worth trying for RMO or INMO or Madhava Mathematics Competition

Mengoli posed the following series to be evaluated:

1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}}+\ldots

Some great mathematicians, including Leibnitz, John Bernoulli and D’Alembert, failed to compute this infinite series. Euler established himself as the best mathematician of Europe (in fact, one of the greatest mathematicians in history) by evaluating this series initially, by a not-so-rigorous method. Later on, he gave alternative and more rigorous ways of getting the same result. 

Question:

Can you show that the series converges and gets an upper limit? Then, try to evaluate the series.

Solution will be posted soon.

Cheers,

Nalin Pithwa.

 

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