<s>[INST] Prove that the sum is always greater than $1$ Prove that $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+...+ \frac{1}{n^2} \ge 1$, for all natural numbers $n$.
I tried to use mathematical inducion but failed and I tried to figure out a short formula for the sum but I couldnt find any.
 [/INST] $$
\sum_{k=n}^{n^2} \frac{1}{k} = \frac{1}{n} + \sum_{k=n+1}^{n^2} \frac{1}{k} \ge \frac{1}{n} + \frac{n^2-(n+1)+1}{n^2} = 1.$$
The final fraction due to there being $n^2-(n+1)+1$ terms, each of which is at least $\frac{1}{n^2}$.
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