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In one of Wikipedia's proofs of the solution to the Basel problem, they state that the $x^2$ coefficient of $$\begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x^2}{\pi^2}\right)\left(1 - \frac{x^2}{4\pi^2}\right)\left(1 - \frac{x^2}{9\pi^2}\right)\cdots \end{align*}$$ can be proven inductively to be $$-\left(\frac{1}{\pi^2}+\frac{1}{4\pi^2}+\frac{1}{9\pi^2}\cdots\right)$$

I do not see any way of proving this by induction. I haven't ever come across this use of induction before actually. Could someone point me in the right direction of how to prove this using induction? I don't even know what the inductive hypothesis would be.

I can see this will certainly be the case though as the only way to have $x^2$ terms is by having each $x^2$ term in each bracket only multiplied together by the $1$'s inside the other brackets.

Thanks for your help.

Hari Ramakrishnan Sudhakar
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A-Level Student
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    I would start by proving using induction $$(1-x^2/\pi^2)(1-x^2/4\pi^2)...(1-x^2/n\pi^2)=-(1/\pi^2+1/4\pi^2..+1/n\pi^2)$$ – Hari Ramakrishnan Sudhakar Jan 01 '21 at 11:42
  • @AlbusDumbledore I see, thanks. But does induction extend to the the case when $n$ is infinity? – A-Level Student Jan 01 '21 at 11:44
  • @AlbusDumbledore eg see here: https://math.stackexchange.com/questions/98093/why-doesnt-induction-extend-to-infinity-re-fourier-series#:~:text=Mathematical%20induction%20affords%2C%20more%20than,is%20distinguished%20from%20the%20infinite.&text=But%20even%20then%2C%20induction%20on,of%20continuous%20functions%20is%20continuous. – A-Level Student Jan 01 '21 at 11:45
  • @Alevel learner in this case i really dont see why induction wouldnt work /directly imply the result( i may be wrong).By the way this is a nice result! – Hari Ramakrishnan Sudhakar Jan 01 '21 at 11:48
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    In this case induction actually shows that ALL "partial products" works, which is exactly what "..." means in this case. The meaning of "..." is pretty nuanced in different situations. – cr001 Jan 01 '21 at 11:51

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There's a footnote in the Wikipedia page, did you read it? It's a formula for the coefficient of $x^2$ of the partial product. That's what can be proved by induction

jjagmath
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