Integration/Reimann Sum Problem

[bbl]

I have been stuck on this problem. I know that I can somehow apply the concept Reimann sum to this but I'm not exactly sure how and where to begin. Any help is appreciated. Thank you!

 

[BigBeachBanana]

Rewrite as summation and factor out n:
[math]\lim_{n\to \infty}\sum_{k=1}^{n}\frac{1}{\sqrt{n}\sqrt{n+k}}= \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n}\frac{1}{\sqrt{1+\frac{k}{n}}} [/math]

 

[bbl]

Rewrite as summation and factor out n:
[math]\lim_{n\to \infty}\sum_{k=1}^{n}\frac{1}{\sqrt{n}\sqrt{n+k}}= \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n}\frac{1}{\sqrt{1+\frac{k}{n}}} [/math]

Ooooh I see it now! Then I can just use integration afterward. Thank you!

 

[Steven G]

Ooooh I see it now! Then I can just use integration afterward. Thank you!

Is it that simple? Integrals are for continuous functions. Your function looks to be discrete.

 

[bbl]

Is it that simple? Integrals are for continuous functions. Your function looks to be discrete.

Oh right. Then, what can I do after factoring out n?

 

[BigBeachBanana]

Note that
[math] \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n}\frac{1}{\sqrt{1+\frac{k}{n}}} [/math]is a Riemann Sum for [imath]f:[0,1] \to \R[/imath], where [imath]f(x)=\frac{1}{\sqrt{1+x}}[/imath]

 

[bbl]

Note that
[math] \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n}\frac{1}{\sqrt{1+\frac{k}{n}}} [/math]is a Riemann Sum for [imath]f:[0,1] \to \R[/imath], where [imath]f(x)=\frac{1}{\sqrt{1+x}}[/imath]

This is what I did. I forgot to clarify but thank you again!