Why were the people of the $18$th century interested in the Basel problem?
(The Basel problem asks for the value of $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}$).
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$\begingroup$ it has number theoretical uses and it was a challenge that Bernoulli brothers failed, where Euler's amazing infinite factorisation of sine series found the result. The $\sum \frac{1}{k^2+k}$ was a known result, but no progress was being made with $\sum \frac{1}{k^2}$. en.wikipedia.org/wiki/Basel_problem $\endgroup$– jimjimCommented Oct 20, 2018 at 12:22
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$\quad$ Because the divergence of the famous harmonic series has already been known for centuries by then, so, in time, they naturally became curious in the convergence and value of its generalization, especially in light of the infinitesimal calculus introduced independently by Newton and Leibniz in the century before.
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$\begingroup$ I am somewhat sceptical of this. My impression is people rarely just consider generalisations of various things unless there is a good reason. Do you perhaps know of any book where I could read more about this? $\endgroup$– TimotejCommented Apr 30, 2014 at 1:45
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$\begingroup$ @Timotej: As I said, the reason or motivation behind this quest was probably to test or exercise the newly invented calculus, dealing with series and integrals (the former being discrete sums, and the latter continuous). $\endgroup$– LucianCommented Apr 30, 2014 at 7:41
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$\begingroup$ A mathematically oriented, but still historical perspective on this problem, including Euler's first solution to it, can be found in Havil's Gamma book. $\endgroup$– mdsCommented May 24, 2018 at 13:47