<s>[INST] Is there a straight forward way to prove that $\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{31} > 3$ Using brute force, it is straight forward to calculate that  $\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{31} > 3$
Is there a more elegant way to demonstrate this?
What if I want to find the smallest $n$ such that $\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} > 4$
Is there a standard way to solve for $n$ without using brute force?
 [/INST] Claim: $$\sum_{i=1}^{3^n}\frac{1}{i} \geq \frac{7}{6}+\frac{2}{3}n$$ 
whenever $n\geq2$.
Proof: By Mathematical Induction. 
Take $n=3$, then instantly $1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{27}\geq \frac{7}{6}+2 >  3$. 
Hope this is not a brutal way and this can be easily generalized. 
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