MHB Series Convergence and Divergence II

[ardentmed]

Hey guys,

I have a few more questions for the problem set I'm working on at the moment:

I'm unsure about b in particular. I compared the series to 1/(n^3/2), which makes it absolutely convergent by the p-test and comparison test. Do I still have to perform any other tests to confirm absolute divergence?

Also, for a, is the integral test another feasible test aside from the alternating series test? With the (-1) taken into account, the function would not be over positive values and thus fails the integral test. However, the absolute of the function would most definitely work. If so, the integral test concludes divergence, but the alternating series test is convergent. Thus, the series is conditionally convergent.

As for c, it's obviously divergent since (n+1)/e is infinity.
Thanks again.

 

[Evgeny.Makarov]

I'm unsure about b in particular. I compared the series to 1/(n^3/2), which makes it absolutely convergent by the p-test and comparison test. Do I still have to perform any other tests to confirm absolute divergence?

No, you don't need other tests.

Also, for a, is the integral test another feasible test aside from the alternating series test? With the (-1) taken into account, the function would not be over positive values and thus fails the integral test. However, the absolute of the function would most definitely work. If so, the integral test concludes divergence, but the alternating series test is convergent. Thus, the series is conditionally convergent.

You are correct, the integral test establishes that the series is not absolutely convergent.

As for c, it's obviously divergent since (n+1)/e is infinity.

Yes, if by "obviously" you mean "by ratio test" and by "is" you mean "tends to".