Take a conditionally convergent series $\sum a_n$. Consider $\sum (-1)^n a_n$. Can we find an example where this second series diverges? More in general, can we always change the signs (following whatever pattern, not just alternating) of a conditionally convergent series' terms so that it diverges?
Changing signs to a conditionally convergent series
sequences-and-series
Best Answer
First, sure, with $a_n=(-1)^n/n$ flipping the signs in an alternating pattern does what you want.
Yes. For a conditionally convergent but not convergent infinite sum of real numbers $a_n$, take a sign of $-1$ for $a_n<0$ and $+1$ for $a_n\ge 0$. The resulting altered sum is divergent.
For complex $a_n$, fooling around with real and complex parts achieves a similar effect, since not both the real and complex parts can be absolutely convergent. Arrange the $\pm 1$'s so that whichever-it-is consists of all non-negative terms...