Prove inequality of series sum by induction

Question

How should i prove by induction that:

$ 1 + 2^{(-2)} + 3^{-2} + ...+n^{(-2)} \le 2-n^{-1} $

I haven't had inequalities with sum of series at the class about induction so i can't really wrap my head around it.

Answer 1

Induction is induction.

Base case: $1^{-2} = 1 \le 2- 1^{-1} = 1$.

Induction case:

If $1 + 2^{-2}+ 3^{-2} + ....... + n^{-2} \le 2-n^{-1}$ then

$1 + 2^{-2}+ 3^{-2} + ....... + n^{-2} + (n+1)^{-2} \le 2-n^{-1} + (n+1)^{-2}$.

So all we have to do is prove $ 2-n^{-1} + (n+1)^{-2} \le 2- (n+1)^{-1}$.

Can we do that?

All we have to do is calculate that

$2 - \frac 1n + \frac 1{(n+1)^2} =$

$2-(\frac 1n - \frac 1{(n+1)^2}) =$

$2 - (\frac {(n+1)^2- n}{n(n+1)^2} )=$

$2 - (\frac {n^2 +n + 1}{n(n+1)^2} )<$

$2 - (\frac {n^2 + n}{n(n+1)^2}) =$

$2- \frac {n+1}{(n+1)^2} =$

$2-\frac 1{n+1}= 2-(n+1)^{-1}$.

And that's it.

So $1 + 2^{-2}+ 3^{-2} + ....... + n^{-2} + (n+1)^{-2} \le 2-n^{-1} + (n+1)^{-2}< 2-(n+1)^{-1}$