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The question we need to prove is $$1.222\le 1+3^{-2}+5^{-2}+\cdots\le 1.252.$$ I know how to prove the first part $$1+3^{-2}+...+43^{-2}\ge 1.222.$$ But I do not know how to prove the second part.

If I upper bound it using $1.222 + 45^{-2} + \text{integration from $45$ to $\infty$}$, I found that it is about $1.277777$ which is too large.

How should I do this question? Thank you very much.

Tianlalu
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user614642
  • 45
  • The middle term is $$ \int_{0}^{1}\frac{\log x}{x^2-1},dx$$ and it can be approximated through $$ \int_{0}^{1}\frac{\log x}{x^2-1}x^4(1-x)^4,dx.$$ The exercise is equivalent to finding a reasonable approximation for $\pi^2$. – Jack D'Aurizio Nov 11 '18 at 18:28
  • Beuker integral gives that the middle term is at most $3\cdot 10^{-4}$ apart from $\frac{145049}{117600}$. By truncating the continued fraction of this rational number, we have that the middle term is at most $4\cdot 10^{-4}$ apart from $\frac{37}{30}$. This is both an accurate and simple approximation. – Jack D'Aurizio Nov 11 '18 at 18:37

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