Riemann discovered that a conditionally convergent series, through rearrangement of it's terms, can be made to converge to any value.
But, if $S$ is a divergent alternating series, through rearrangement of terms of $S$,can we make it convergent to some value?
Thank you!
Best Answer
Maybe or maybe not. If you follow through the proof that rearranging the conditionally convergent series can make the sum anything you want, an important point is that the sum of the positive terms diverges to $+\infty$ and the sum of the negative terms diverges to $-\infty$. If that happens, you can use the same style of rearrangement. If not, consider the series $$a_n=\begin {cases} 2^{-n}&n \text { odd}\\ -1& n \text { even} \end {cases}$$ This is a diverging alternating series, but you cannot make it converge to anything. If you insist that the terms go to zero in absolute value (as required in the alternating series theorem) you can use $$a_n=\begin {cases} 2^{-n}&n \text { odd}\\ -1/n& n \text { even} \end {cases}$$