Hi guys.

This might be a dumb question, I don't know. We're having mid-terms tomorrow for Calc. I've got Fourier Series down, Taylor polynomials and series, Power series, the remainder theorem, Sigma notation and yadayada.

But I don't feel like I have convergence down. In particular, I'm not quite sure how to determine when a value converges conditionally.

I know I've asked this question before, and I believe someone guided me to WolframMath. But if someone could demonstrate an example of conditional convergence compared to absolute convergence, it might really clear things up for me.

Maybe I'm overthinking it, I don't know.

Thanks if you have the time.

 

An example of conditional convergence is here:
http://en.wikipedia.org/wiki/Conditional_convergence

You should also follow the "see also" stuff to get a better idea.

 

Conditionally convergent means that the series is convergent but not absolutely convergent. Absolute convergence means that the the absolute value of An is convergent. Alot of the time alternating series will converge by the alternating series test, but the absolute value diverges because of p series.

[(-1)^n+1] / sqrt n is a good example.

1/ sqrt n diverges by p series, p < 1

[(-1)^n+1] / sqrt n converges because it is alternating, decreasing, and limit at infinity goes to 0



is that what was confusing you?

 

Another way to define a convergent series is that it is one that converges, but the series made up of the absolute values of the terms in the original series diverges.

A classic example (and the one given in the wikipedia article) is \(\sum\) (-1)\(^{n + 1}\)/n, where n \(\geq\) 1.