Rudin's principles of mathematical analysis

[ehrenfest]

Homework Statement

Does anyone have this book? I have some questions about chapter 3.

Homework Equations

The Attempt at a Solution

 

[quasar987]

Try to find an expression for the Nth partial sum.

This is an example of what is called a "telescoping series".

 

[ehrenfest]

Try to find an expression for the Nth partial sum.

This is an example of what is called a "telescoping series".

Yes I figured that out before you posted and deleted that part of the post because it was embarrassing.

What about $\sum_n \frac{\sqrt{n+1}-\sqrt{n}}{n}$. This definitely does not telescope. But both 1/n and \sqrt{n+1}-\sqrt{n} diverge so it is pretty clear that their product will. But what test should I use? The ratio test is too hard too evaluate. The root test is even harder to evaluate. What series can I compare it to?

 

[NateTG]

What about $\sum_n \frac{\sqrt{n+1}-\sqrt{n}}{n}$

Hmm, is any term in that series larger than:
$$\frac{1}{2n^{\frac{3}{2}}}$$

(This falls out easily with a little algebra.)

 

[ehrenfest]

Hmm, is any term in that series larger than:
$$\frac{1}{2n^{\frac{3}{2}}}$$

No (but I am not sure why you have the 2 there).

So, does anyone have the book?