Prove by mathematical induction.

Question

TO BE PROVED: $$ \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\dots \frac{1}{n^2}<2 $$
Please prove this by mathematical induction only. My approach: I already proved it through graphs but i have to prove it through mathematical induction. Prove this equation by mathematical induction that $\frac{1}{1} + \frac{1}{2^2} + \frac{1}{3^2} + .... + \frac{1}{n^2} < 2 $ ?

Answer 1

First, let's make the claim stronger - $\displaystyle \forall n\in\Bbb N:\ 1+\frac{1}{2^2}+\frac{1}{3^2}+\dots+\frac{1}{n^2}\le2-\frac{1}{n}<2$

Base case - $n=1$ is trivial.

Step: assume the claim holds for $k$ and prove for $k+1$ $$\underbrace{1+\frac{1}{2^2}+\dots+\frac{1}{k^2}}_{\le 2-\frac{1}{k}\text{ by induction assumption}}+\frac{1}{(k+1)^2}\le 2-\frac{1}{k}+\frac{1}{(k+1)^2}\le 2-\frac{1}{k}+\frac{1}{k(k+1)}\\=2-\frac{1}{k}+\frac{1}{k}-\frac{1}{k+1}=2-\frac{1}{k+1}<2$$

Q.E.D