I am to use mathematical induction to prove:
$$\sum_{i=1}^n\frac{1}{i^2}<2 - \frac{1}{n}$$
my base case is n = 3:
LHS: $\frac{1}{1}+\frac{1}{4}+\frac{1}{9}= \frac{49}{36}$
RHS: $2-\frac{1}{3}=\frac{5}{3}$
base case holds true.
assume $n = k$
$$2-\frac{1}{k}$$
now show $k+1$
$$2-\frac{1}{k+1}$$
Inductive Step:
$$2-\frac{1}{k}+ \frac{1}{{k+1}^2}$$
Now I am stuck I have no clue what math I should be doing from here. I thought about trying to FOIL the $\frac{1}{{k+1}^2}$ but that didn't really give me anything I could work with. Any tips? Or general pointers when doing this stuff? Thanks guys!
Hint: Your inductive step will work if you can show $$2-\frac1k +\frac1{(k+1)^2} \le 2-\frac1{(k+1)}$$ so consider whether $$\left( 2-\frac1{(k+1)}\right) - \left( 2-\frac1k +\frac1{(k+1)^2} \right)$$ is positive, zero or negative.