# Rudin's principles of mathematical analysis

• Mathematica

## Main Question or Discussion Point

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

2. Homework Equations

3. The Attempt at a Solution

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quasar987
Homework Helper
Gold Member
Try to find an expression for the Nth partial sum.

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

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?

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NateTG
Homework Helper
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.)

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?