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In my answer to this question I proved the fact that $$\sum_{r=-\infty}^{\infty}\frac{1}{64r^4+1}=\frac{\pi}{4}\frac{1+\mathrm{sinh}(\pi/2)}{\mathrm{cosh}(\pi/2)}$$ using quite a non-advanced method*; nothing like Fourier series or anything more than a high school student studying mathematics would understand was in it.

I was just wondering, are there any other (hopefully relatively 'low-tech') methods that could be used to prove the above identity, other than using residue calculus?

Thank you for your help.

*(To clarify: although Weierstrass products are definitely too advanced for me and most high school students and I have no idea how to derive them, the Weierstrass products for $\sin x$ and $\cos x$ are very intuitive, and can definitely be understood by high school students on a non-rigorous level (but, as I said, the derivation of it is certainly very advanced). I don't think that any other ideas that I used in my answer are very advanced; after all, I really just use some high-school level trigonometry, complex numbers and calculus. If you think that the method should not be called non-advanced, then please ignore my description of it being non-advanced and pretend you never read it :) )

A-Level Student
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  • How do you suppose the Weierstrass product for the sine function is derived? You didn't see that in high school calculus. – saulspatz Apr 05 '21 at 15:42
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    You call the approach you presented "quite a non-advanced method." If that is considered non-advanced, then surely use of Fourier Series or Contour Integration would be considered non-advanced also. – Mark Viola Apr 05 '21 at 15:57
  • @saulspatz Of course; I've edited my post to clarify that point. The derivation is certainly way beyond me, but the result itself is very intuitive on a basic, non-rigorous level. – A-Level Student Apr 05 '21 at 17:14
  • @MarkViola I didn't really use any ideas that are too advanced for high school students other than the Weierstrass products; the rest is just high school level calculus, complex numbers and trigonometry, don't you think? – A-Level Student Apr 05 '21 at 17:16
  • The product representations are sufficiently advanced to warrant the entire development likewise. – Mark Viola Apr 05 '21 at 19:41
  • @MarkViola I see. Do you think I should delete the words non-advanced then? – A-Level Student Apr 05 '21 at 19:51
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    My point is that Product representations are more advanced than Fourier Series and Contour Integration. You're excluding use of these yet used a more advanced approach yourself. – Mark Viola Apr 05 '21 at 20:35
  • Let us continue this discussion in chat. – A-Level Student Apr 05 '21 at 21:15
  • These sums require complex analysis in an essential way. Ramanujan would have summed it using partial fractions but the rigorous backing for his method is again complex analysis. See related https://mathoverflow.net/q/173356/15540 and https://math.stackexchange.com/q/898955/72031 for the gymnastics done by Ramanujan – Paramanand Singh Apr 06 '21 at 02:53

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