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Compare positive denominators to prove a strict lower bound

Given ∑r=1n1/(r+1+r)=n+1−1\sum_{r=1}^n1/(\sqrt{r+1}+\sqrt r)=\sqrt{n+1}-1, deduce ∑r=1n1/r>2(n+1−1)\sum_{r=1}^n1/\sqrt r>2(\sqrt{n+1}-1). marks

[2013/JJC/I/3(modified)]

CJC · Review problems · Question 9

Compare the denominators before taking reciprocals; all quantities are positive.

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