$1/4 + 1/ 9 + ... 1/n^2< 1$ induction

Question

I have been trying this sum for long and do not know how to proceed.

Q. Prove using induction that $$\frac1 4 + \frac1 9 + ... + \frac 1 {n^2} < 1$$

A. By induction.

Let $$P(n) = \frac1 4 + \frac1 9 + ... + \frac 1 {n^2} < 1$$

Base Case, $P(2):\frac 1 4 < 1$, True.

Inductive Step: Assuming $P(n)$ is true

$$P(n+1) = \frac1 4 + \frac1 9 + ... + \frac 1 {n^2} + \frac 1 {n^2 + 2n +1}< 1$$

I am stuck now and don't know how to proceed. Any help would be appreciated.

Answer 1

This is one of those cases where proving something is most easily done by proving something stronger. For instance, you might be able to prove by induction that $$ \frac14 + \frac19 + \cdots +\frac1{n^2} < 1-\frac1n $$ which would directly imply $$ \frac14 + \frac19 + \cdots +\frac1{n^2} < 1 $$ The big difference in the stronger version is that you actually get some wriggle room when you do the induction step, which makes it possible to actually complete the induction.