Integral test for series

[fred2028]

Hi, I have learned that in order to be able to use the integral test on a series, the terms must be decreasing and positive. Assuming we have a series that goes from 1 to infinity, my questions are:

1) Do the terms have to be positive completely in the interval or do they just have to be positive as we approach infinity? So would 5 + 4 - 3 + 2 + ... work?
2) Do the terms have to be decreasing completely in the interval or do they just have to be decreasing as we approach infinity? So would 1 + 2 + 10 + 3 + 2 + ... work?

My second question is that, given that we have numerous different divergence tests, how do we know which 1 is the appropriate one? I am doing homework atm and occasionally the solutions manual comes up with a solution that I wouldn't have considered.

And I guess lastly, what is the point of learning all about divergence and convergence? Is this leading up to something?

Thanks!

 

[fasteddie65]

Hi, I have learned that in order to be able to use the integral test on a series, the terms must be decreasing and positive. Assuming we have a series that goes from 1 to infinity, my questions are:

1) Do the terms have to be positive completely in the interval or do they just have to be positive as we approach infinity? So would 5 + 4 - 3 + 2 + ... work?

In order to apply the integral test, the series (in terms of n) must be rewritten as a function in terms of x.
For example, the series ? 1/n^2 can be rewritten as ? 1/x^2. Then you integrate and see if the integral has a limit.

2) Do the terms have to be decreasing completely in the interval or do they just have to be decreasing as we approach infinity? So would 1 + 2 + 10 + 3 + 2 + ... work?

If it says decreasing, it means decreasing.

My second question is that, given that we have numerous different divergence tests, how do we know which 1 is the appropriate one? I am doing homework atm and occasionally the solutions manual comes up with a solution that I wouldn't have considered.

And I guess lastly, what is the point of learning all about divergence and convergence? Is this leading up to something?

Thanks!