Hello all,

I have a big ol' ugly mid-term tomorrow on series convergence. Tonight I've been clunking through all the definitions and endless theorems and have become curious about a couple of things.

The integral test for convergence works in the case of functions that are positive, decreasing, and continuous, right? Well, my question here is why not alternating, or discontinuous? Hey, at least I get the decreasing part...right? Anyway:

1) Take the case of a function that is asymptotic at some number, let it be c. And c is an integer, as they often are. In that case, why could we not evaluate the improper integral from 0 to c, and c to infinity? We do allow an asymptote, and an infinite interval in many functions for which we evaluate integrals, do we not?

2) In the case of an alternating series, I believe that the series sum is the absolute value of sum a of n as n goes to infinity, right? (Please correct me if i am wrong, and forgive my lack of latex chops) Hence, my question is why would that sum not be the abs value of the function's integral?

OK, that's it for now.

Thanks!

1) It depends on whether the function approaches that value from both sides. In other words, the limit has to exist.

2) No, the series sum is not the same as the sum of the absolute values.

HOWEVER

If you can prove that the sum of the absolute values CONVERGES, then so does the alternating series. In fact, so does any series with the same absolute values (it doesn't even necessarily have to be alternating).

It's relatively easy to understand using the comparison test.

It should be obvious that any number is no greater than its absolute value.

I.e. \(\displaystyle a_n \leq |a_n|\).

Therefore, the sum of numbers is never any greater than the sum of their absolute values.

I.e. \(\displaystyle \sum_n{a_n} \leq \sum_n{|a_n|}\).

It should make sense that if the "larger" series converges to a number, than anything "smaller" must also converge to a (smaller) number.

So that means, if \(\displaystyle \sum_n{|a_n|}\) is convergent, then so is \(\displaystyle \sum_n{a_n}\). If you prove convergence this way, then you say the series is ABSOLUTELY CONVERGENT.

Of course, if you prove that the sum of the absolute values is divergent, that doesn't mean that the original series is divergent. You would have to use some other test (e.g. Root Test, Lagranges Alternating Series Test). If a series is not absolutely convergent, but can still be proven to be convergent, then you say it is CONDITIONALLY CONVERGENT.