So in Euler's solution of the Basel problem he takes $\sin x= x\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{4\pi^2}\right)\cdots$

well, i was playing around with it and put $ x=\frac{\pi}{2}$

After manipulating quite a bit, $$\frac{2}{\pi}=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{(2*2)^2}\right)\cdots\left(1-\frac{1}{(2n)^2}\right)\cdots$$ after simplifying a bit$$\frac{2}{\pi}=\left(\frac{3}{4}\right)\left(\frac{15}{16}\right)\left(\frac{35}{36}\right)\left(\frac{99}{100}\right)\cdots$$

I tried evaluating LHS by hand, and the partial products slowly do decrease towards RHS, and we know that they will not decrease forever because $$\lim_{n\to\infty} \frac{n^2-1}{n^2}=1$$ so can anyone help by writing a program which can do partial sum up to high values of n and tell me if its true. PS.I am in 10th standard do if im wrong anywhere please dont explain in too complicated mathematics. thanks


marked as duplicate by Guy Fsone, Sahiba Arora, Shailesh, JonMark Perry, Claude Leibovici Jan 16 '18 at 9:29

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  • $\begingroup$ $\sin x$ cannot equal $\pi/2,$ which is greater than 1. $\endgroup$ – Aurel Jan 15 '18 at 18:19
  • $\begingroup$ You actually put $x=\pi/2$, right? $\endgroup$ – David C. Ullrich Jan 15 '18 at 18:20
  • $\begingroup$ 'Twas a typo, my bad sire. $\endgroup$ – user167920 Jan 15 '18 at 18:29
  • 2
    $\begingroup$ It's a known result, Wallis product, though Wallis didn't prove it that way (predictably, since he did so in 1655, long before Euler's results). $\endgroup$ – Professor Vector Jan 15 '18 at 18:30
  • $\begingroup$ well, i couldn't find anything by "2/pi infinite product identity" on google so i posted it here. thank you for directing me. $\endgroup$ – user167920 Jan 15 '18 at 18:34

$\dfrac2{\pi} \approx 0.6366198$ is the correct limit of the product

As an illustration (not a proof) using R:

maxn  <- 2^15
plot(exp(cumsum(log(1 - (2*(1:maxn))^(-2)))), log="x")
abline(h=2/pi, col="red")


enter image description here

  • $\begingroup$ thank you, for the program mister henry $\endgroup$ – user167920 Jan 15 '18 at 18:54

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