<s>[INST] Prove inequality of series sum by induction 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.
 [/INST] 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 $1 + 2^{-2}+ 3^{-2} + ....... + n^{-2} + (n+1)^{-2} \le 2-n^{-1} + (n+1)^{-2}<  2-(n+1)^{-1}$
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