Quick series convergence question

Click For Summary

Discussion Overview

The discussion revolves around determining the convergence or divergence of the series \(\sum \left(\sqrt{n+1}-\sqrt{n}\right)\). Participants explore various methods for analysis, including the ratio test, root test, and comparison tests.

Discussion Character

  • Exploratory, Homework-related, Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about how to show the convergence or divergence of the series, noting that the ratio test appears inconclusive and seeking alternatives.
  • Another participant suggests trying the integral comparison test or multiplying by the conjugate to clarify the series for comparison.
  • A later reply indicates that multiplying by the conjugate did not yield a helpful result, and the limit comparison tests were also unsuccessful.
  • One participant later claims to have resolved the issue, stating that adding the terms leads to divergence.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the method for determining convergence or divergence, and multiple approaches are discussed without resolution.

Contextual Notes

Some methods mentioned, such as the ratio test and limit comparison tests, are noted as inconclusive or unhelpful, indicating potential limitations in the approaches discussed.

DE7
Messages
8
Reaction score
0
Hey all, it's been a while since I've done series and I have a quick question. How would I show the convergence or divergence of [itex]\sum \left(\sqrt{n+1}-\sqrt{n}\right)[/itex]? The ratio test is inconclusive I think, and I'm not sure how I would go about doing the root test. Or is there a series I could compare it to for the comparison test? Anyways, thanks, just need a little refresher.
 
Physics news on Phys.org
DE7 said:
Hey all, it's been a while since I've done series and I have a quick question. How would I show the convergence or divergence of [itex]\sum \left(\sqrt{n+1}-\sqrt{n}\right)[/itex]? The ratio test is inconclusive I think, and I'm not sure how I would go about doing the root test. Or is there a series I could compare it to for the comparison test? Anyways, thanks, just need a little refresher.

You could try the integral comparison test. Or you could try multiplying by the conjugate, this should make the series to compare with more transparent.
 
Still not sure what to do...multiplying by the conjugate gives me something that doesn't seem so helpful to me, as I still can't find a comparison series. The limit comparison tests don't seem to be working either. Any more hints please?
 
Never mind...got it.
 
Add them together you'll get f(n+1)-1, it's divergence.
 

Similar threads

Undergrad Infinite Series of Infinite Series
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Solving a power series
  • · Replies 3 ·
Replies
3
Views
5K
High School Difficulty with understanding whether 1/n converges or diverges
  • · Replies 11 ·
Replies
11
Views
6K
Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
5K
Undergrad Infinite Series - Divergence of 1/n question
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Understanding the nth-term test for Divergence
  • · Replies 17 ·
Replies
17
Views
6K
Graduate How Can We Ensure Convergence in Function Approximations Beyond Taylor Series?
  • · Replies 23 ·
Replies
23
Views
4K
Undergrad Convergence of an infinite series
  • · Replies 15 ·
Replies
15
Views
3K
MHB Does the Comparison Test Determine Convergence or Divergence of Series?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Convergence and divergence of series and sequences
  • · Replies 7 ·
Replies
7
Views
4K