TITLE: Prove $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{n}} > 2\:(\sqrt{n+1} − 1)$
QUESTION [11 upvotes]: Basically, I'm trying to prove (by induction) that:
$$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{n}} > 2\:(\sqrt{n+1} − 1)$$
I know to begin, we should use a base case.  In this case, I'll use $1$.  So we have:
$$1 > 2\:(1+1-1) = 1>-2$$
Which works out.
My problem is the next step.  What comes after this?  Thanks!

REPLY [6 votes]: Let $f(x) = \frac{1}{\sqrt{x}}$. This is a decreasing function on $\mathbb{R}^+_*$, hence
$$ f(k) > \int_k^{k+1} f(x) \, \mathrm{d}x. $$
Summing for $k=1, \dots, n$ give
$$ f(1) + f(2) + \dots + f(n) > \int_1^{n+1} f(x) \, \mathrm{d}x,$$
which is exactly the desired inequality.