# Prove a Sum is Larger than a Root

1. Feb 16, 2012

### hammonjj

1. The problem statement, all variables and given/known data
Let nεZ>0. Prove that:

Ʃ1/√(i) ≥ √(n)

3. The attempt at a solution
I'm not sure where to begin, this feels like it should be an induction problem, but I'm not sure how to show that √(n) + 1/√(i+1) ≥ √(n+1). There doesn't seem to be any obvious algebra that would simplify this into what I need it to be.

2. Feb 17, 2012

### Some Pig

Statement correct for n=1.
For $n\ge2,$
$$\sum_{i=1}^n\frac1{\sqrt i}\ge\int_1^{n+1}\frac1{\sqrt x}\ dx =2\sqrt{n+1}-2\ge\sqrt n.$$

3. Feb 17, 2012

### dirk_mec1

No, you have to show that:

√(n) + 1/√(n+1) ≥ √(n+1).

In your last step of the induction proces.