Convergence of Series & Sequences: Tests Explained

[STLCards002]

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.

 

[quasar987]

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.

 

[HallsofIvy]

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.

 

[STLCards002]

Well can you list the various tests that I can try to apply

I understand the a_{}n+1/a_{}n 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...

 

[jacobrhcp]

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.