$1 + 1/4 + 1/9 + 1/16 + .... + 1/n^2 < 2 - \frac{1}{n}$ for all $n \ge 2$ , $n \in N$.

Question

I proved this inequality using mathematical induction. I was wondering if it can be proved in any other way. Please help me.

Answer 1

You can use the fact that\begin{align}\require{cancel}1+\frac1{2^2}+\cdots+\frac1{n^2}&<1+\frac1{1\times2}+\frac1{2\times3}+\cdots+\frac1{(n-1)\times n}\\&=1+1-\cancel{\frac12}+\cancel{\frac12}-\cancel{\frac13}+\cancel{\frac13}+\cdots-\frac1n\\&=2-\frac1n.\end{align}

Answer 2

For example you can prove this by integration. For $n > 1$ $$ \sum\limits_{i = 1}^{n} \frac{1}{i^2} < 1 + \int\limits_{1}^n\frac{dx}{x^2} = 1 + (- \frac{1}{x})|_1^n = 2 - \frac{1}{n} $$