<s>[INST] Using mathematical induction to prove $\frac{1}1+\frac{1}4+\frac{1}9+\cdots+\frac{1}{n^2}<\frac{4n}{2n+1}$ This induction problem is giving me a pretty hard time: 

$$\frac{1}1+\frac{1}4+\frac{1}9+\cdots+\frac{1}{n^2}<\frac{4n}{2n+1}$$

I am struggling because my math teacher explained us that in this case ($n^2$) when we prove that $n+1$ satisfies the property we have to write it like this: $$ \frac{1}1+\frac{1}4+\frac{1}9+\cdots+  \frac{1}{n^2} +\frac{1}{(n+1)^2} < \frac{4(n+1)}{(2(k+1)+1)}$$ 
and I probably got lost in the process.
 [/INST] Hint: What is $\frac{4(n+1)}{2(n+1)+1}-\frac{4n}{2n+1}$? (What is the corresponding difference on the other side of the inequality when you move from $n$ to $n+1$?)
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