Convergence of Series & Sequences: Tests Explained

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Discussion Overview

The discussion centers around the convergence of series and sequences, specifically exploring various tests used to determine convergence. Participants express challenges in understanding the reasoning behind these tests and seek clarification on their application.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant requests an explanation of how series and sequences converge and the tests associated with them.
  • Another participant suggests that there are no definitive rules for which test to apply, emphasizing the development of intuition over time.
  • A different participant reiterates that the process involves trial and error, trying different tests until one is successful.
  • One participant asks for a list of various convergence tests and expresses confusion about the reasoning behind them.
  • Another participant provides a link to a resource on convergence tests and suggests looking for proofs in calculus books, while also cautioning that understanding the proofs may be challenging for novices.

Areas of Agreement / Disagreement

Participants generally agree that there is no single method for determining which convergence test to apply, and multiple views remain on the best approach to understanding these tests.

Contextual Notes

Participants express uncertainty about the reasoning behind specific convergence tests and the applicability of those tests to different series and sequences.

Who May Find This Useful

Individuals interested in the convergence of series and sequences, particularly those seeking to understand the various tests and their applications.

STLCards002
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Can somebody please explain to me:
How both series and sequences converge, and the various tests to find out.

I've tried searching but it seems impossible to get any explanations as to why you do the specific test.
 
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There isn't. You just try them one by one. Except after a while you develop an intuition as to which test will work on which series.

It's like integration. After a while you see whether integration by parts will be fruitful or not.
 
There are no general rules that say "apply this test to these kinds of problems". You try one and if it doesn't work you try another.
 
Well can you list the various tests that I can try to apply

I understand the a[tex]_{}n+1[/tex]/a[tex]_{}n[/tex] test where if the ratio is > 1 and it is bounded about then the sequence converges and < 1 and bounded below then it converges. But I don't understand the reasoning behind the other ones.

Those should be subscripts sorry...
 
Last edited:
hm... I think this should help you out.

http://en.wikipedia.org/wiki/Convergence_tests

edit: if you are looking for the proofs of these tests you should look in a calculus book or search a little harder on the internet. You can also try them yourselves, in which case you should remember that most of the proofs are based on the geometric series. But I think as a complete novice to the subject, trying to figure out these proofs yourselves might be a little too much of a good thing.
 
Last edited:

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