Problem with proving an inequality by mathematical induction

Question

I wonder how to solve this problem using mathematical induction:

$1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + ... + \frac{1}{\sqrt{n}} > \sqrt{n}, n\geq2$

I showed true for $n = 2$

Assumed true for $n = k$, the tried to prove true for $n = k + 1$, but I was unable to finish proof.

How should I prove this inequality correctly?

Does this answer your question? Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction — F Nishat, May 06 '21 at 11:04

score 0 · Answer 1 · answered May 06 '21 at 11:02

Let $S(n)=\sum_{j=1}^n\frac {1}{\sqrt j}$ and let $S(n)=D(n)+\sqrt n.$

Then $D(n+1)=S(n+1)-\sqrt {n+1}=$

$=S(n)+\frac {1}{\sqrt {n+1}}-\sqrt {n+1}=$

$=D(n)+\sqrt n+\frac {1}{\sqrt {n+1}}-\sqrt {n+1}=$

$=D(n)+\frac {1}{\sqrt {n+1}}-\frac {1}{\sqrt n\,+\sqrt {n+1}}>D(n).$

So if $D(n)\ge 0$ then $D(n+1)>0.$

1 Answers1