I've gone through countless websites that promise answering "What is conditional convergence?" and instead give me "This is how you find if something is conditionally convergent".
While it is all fine and dandy that I know how to find if something is conditional convergence, what actually is it?
I've graphed severally conditionally convergent series and they seem to not look much different from other series which have absolute convergence. What is the magic thing that makes something converge conditionally? I'm just confused because I don't understand how something can converge but actually not really all the time.
Answer
"Conditional" is a bit of a strange adjective to use. After all, a series either converges or it doesn't: what is conditional about that?
The reason for the word "conditional" is that, given any series which converges but does not converge absolutely, it is possible to rearrange the series (i.e., reorder the terms) in such a way that the series no longer converges.
It is also possible, given any desired value $V$, to find a rearrangement of the series which converges to $V$.
This is known as the Riemann rearrangement theorem.
Note that this phenomenon does not occur with absolutely convergent series. Given any absolutely convergent series, we can rearrange the terms any way we like, and it will still converge to the same value.
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