Convergence of Series & Sequences: Tests Explained

In summary, the conversation discusses the concept of convergence for both series and sequences and the various tests that can be used to determine convergence. The conversation also mentions that there are no general rules for which test to apply and that intuition plays a role. The ratio test is briefly explained, but the reasoning behind the other tests is not understood. The recommendation is made to consult a calculus book or the internet for further understanding.
  • #1
STLCards002
12
0
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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  • #2
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.
 
  • #3
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.
 
  • #4
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...
 
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  • #5
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.
 
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FAQ: Convergence of Series & Sequences: Tests Explained

1. What is the definition of convergence in series and sequences?

Convergence in series and sequences refers to the behavior of a sequence or series as its terms approach a certain value, known as a limit. If the terms of the sequence or series approach a specific value or tend towards a certain pattern, then it is said to converge. On the other hand, if the terms do not approach a specific value or pattern, then it is said to diverge.

2. What are some common tests for convergence in series and sequences?

Some common tests for convergence in series and sequences include the ratio test, the root test, the comparison test, the integral test, and the alternating series test. These tests can be used to determine whether a series or sequence converges or diverges.

3. How is the ratio test used to determine convergence?

The ratio test is used to determine the convergence of a series by comparing the ratio of consecutive terms to the limit of the ratio as n approaches infinity. If the limit is less than 1, the series will converge, and if it is greater than 1, the series will diverge. If the limit is equal to 1, the test is inconclusive and another test must be used.

4. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series that converges regardless of the order in which the terms are added. In other words, the series will converge regardless of whether the terms are added from smallest to largest or largest to smallest. On the other hand, conditional convergence refers to a series that only converges if the terms are added in a specific order. If the terms are added in a different order, the series may diverge.

5. Can a series or sequence converge to a value other than its limit?

No, a series or sequence can only converge to its limit. This means that as the number of terms in the series or sequence approaches infinity, the terms will get closer and closer to the limit. However, it is possible for a series or sequence to oscillate or fluctuate before converging to its limit.

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