Proof for inequality induction

[javi438]

proof for inequality induction...please help!

Homework Statement

Prove for all positive integers n that
$$\sum^{n}_{l=1} l^{-1/2} > 2(\sqrt{n+1} -1)$$

2. The attempt at a solution
$$\sum^{n+1}_{l=1} l^{-1/2} = \sum^{n}_{l=1} l^{-1/2} + (n+1)^{-1/2} > 2(\sqrt{n+1} -1) + (n+1)^{-1/2}$$

please help meee! I'm getting stuck, on how to express $$2(\sqrt{n+1} -1) + (n+1)^{-1/2}$$ in terms of $$2(\sqrt{n+2} -1)$$

 

[Dick]

Are you sure you are supposed to prove this by induction? I'm having a hard time doing it that way. On the other hand there is an easy way to do it by looking at an upper Riemann sum and comparing it with the integral of x^(-1/2).

 

[javi438]

yes I am pretty sureee..because that's what I am learning right nowww

 

[Dick]

Ok, so what you want to do is now show that 2*(sqrt(n+1)-1)+1/sqrt(n+1)>2*(sqrt(n+2)-1). Right? Multiply both sides by sqrt(n+1). The simple sqrt(n+1) radicals cancel on both sides. The sqrt(n+1)sqrt(n+2) remains on the right. Square both sides. It just barely works.