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.