# Thread: infinite series convergence theorems

1. ## infinite series convergence theorems

Let S be an infinite series.
Let S+ be the infinite series which consists of the positive terms in S (the nth term of S+ is 0 if the nth term of S is negative, otherwise the terms of S+ = the terms of S)
Let S- be the infinite series which consists of the negative terms of S.

From my text I have gathered that:

S+ and S- both converge <=> S is absolutely convergent
S+ and S- both diverge <= S is conditionally convergent
only one of S+ and S- converges, the other diverges => S is divergent

But it is not clear in the text whether

S+ and S- both diverge => S is conditionally convergent
only one of S+ and S- converges, the other diverges <= S is divergent

ie. if both S+ and S- diverge, can S diverge?

2. What about $1-2+3-4+5-\cdots$ or $\frac12 +\frac13 - \frac13 + \frac14 + \frac15-\frac15+\cdots$?