2007-10-30 - More on alt-series

Today's nonsense
- Gilson bashing. Apparently two students in Andrew's second-year algebra class have cried because of Gilson's death-ray eyes.

Today we are going to do more on alternate series.

Absolute and Conditional Convergence

If the series Sum of abs(a_{_n}) converges, then the series Sum of a_{_n} converges.
-- If some of the numbers are negative, then that's just going to cancel out in the first place. So if you have a series that has all positive terms, and if all of those terms somehow converge, and the series converges, then if the same series has some negative terms (even if it is not alternating) then that will also converge.

When you have an alternating series, it will not be good enough to say that it converges, you have to specify if it converges absolutely or conditionally. So if you do not specify anything, it means that it's absolutely convergent, but with alternate series there's also something called conditionally convergent that we will now deal with.

Definition of Absolute and Conditional Convergence

1. Series a_{_n} is absolutely convergent if the Series of the absolute value of a_{_n} is convergent.
2. Series a_{_n} is conditionally convergent if the Series a_{_n} converges but Series abs(a_{_n}) is divergent.

ex) Determine if the following either converges absolutely or converges conditionally, or if it diverges.

a) Sum from n=1 to infinity of (-1)^n * 1/n ....
- The original series converges. However, the absolute value of the sequence diverges, so the original one converges conditionally.

b) Series from n=1 to infinity of (-1)^n * ( n^2 / (n+3)^2 )
- Diverges.

c) Series from n=1 to infinity of (-1)^n / (2n+1)!
- Absolutely convergent, but how can we show that the absolute value part is satisfied? We do not yet know the ratio test.
- Compare with n^2 ... and then you know that (2n+1)! > n^2 ... and n^-2 converges by the p-series test or the integral test.

Bonus: is there a way to make a series that has a negative term every third term?