Rudin's principles of mathematical analysis

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Discussion Overview

The discussion revolves around questions and challenges related to chapter 3 of Rudin's "Principles of Mathematical Analysis." Participants are exploring concepts related to series, specifically telescoping series and convergence tests.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant inquires about the book and mentions having questions about chapter 3.
  • Another participant suggests finding an expression for the Nth partial sum of a series, identifying it as a telescoping series.
  • A participant acknowledges understanding the telescoping nature but raises a different series, \(\sum_n \frac{\sqrt{n+1}-\sqrt{n}}{n}\), questioning its convergence and the appropriate test to use.
  • There is a suggestion to compare terms in the series to \(\frac{1}{2n^{\frac{3}{2}}}\), with a participant expressing uncertainty about the reasoning behind the factor of 2.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the convergence of the series in question or the reasoning behind the comparison made. There are multiple competing views regarding the appropriate convergence tests and the nature of the series.

Contextual Notes

Some assumptions about the series and convergence tests remain unaddressed, and the discussions reflect uncertainty in evaluating the series and the rationale behind specific comparisons.

ehrenfest
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Homework Statement


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

Homework Equations


The Attempt at a Solution

 
Last edited:
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Try to find an expression for the Nth partial sum.

This is an example of what is called a "telescoping series".
 
quasar987 said:
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 [itex]\sum_n \frac{\sqrt{n+1}-\sqrt{n}}{n}[/itex]. 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?
 
Last edited:
ehrenfest said:
What about [itex]\sum_n \frac{\sqrt{n+1}-\sqrt{n}}{n}[/itex]

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

(This falls out easily with a little algebra.)
 
NateTG said:
Hmm, is any term in that series larger than:
[tex]\frac{1}{2n^{\frac{3}{2}}}[/tex]

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

So, does anyone have the book?
 

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