Rearrange a convergent alternating series to make it diverge [read desc.]

Question

Can we re-arrange the alternating series $+1 -1 +0.5 -0.5 +\cfrac13 -\cfrac13 ... +\cfrac1m -\cfrac1m$ so it diverges ?

Is that possible? Any possible way?

Answer 1

You are asking whether this series is conditionally convergent or if it is absolutely convergent.

Answer 2

E.g. take positive terms until the partial sum $>1$, then one negative term, then more positive terms until the partial sum $>2$, then another negative term, etc.

Answer 3

It is possible if and only if the series is absolutely divergent. In this case it is, because the harmonic series diverges.

To be more precise:

Theorem: let $u_n$ be a series of real numbers such that $\sum u_n$ converges but $\sum |u_n|$ diverges. Then for any $L\in \mathbb{R}\cup \{-\infty, +\infty\}$ there is a permutation $\phi$ of $\mathbb{N}$ such that $\sum u_{\phi(n)}$ tends to $L$.