# Alternating Series Question

1. Apr 20, 2013

### That Neuron

1. The problem statement, all variables and given/known data

The problem contained five answer choices, of which I the answerer was to find one that fit the criteria of the question. The question was: "Which series of the following terms would be convergent?".

2. Relevant equations

$\Sigma$1 + (-1)n / n.

3. The attempt at a solution

I find this very confusing simply because whilst separating the series into two separate series, $\Sigma$1 and $\Sigma$(-1)n / n, The second series converges (yes I was surprised also) by the alternating series test. Originally, I was dumbfounded because of the absolute value test, so I suppose the series is conditionally convergent. Anyways, if $\Sigma$(-1)n / n is convergent and $\Sigma$1 is divergent, 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.... + 1 .... +1 (you get the point) Then how can a finite number (the second alternating series) affect convergence? Also, the limit test kind of rules convergence out for this one. Ha.

I may be missing something deceptively simple, so if anyone can help me out here that'd be great!

2. Apr 20, 2013

### jbunniii

Do you mean this series:
$$\sum_{n=1}^{\infty}\left(1 + \frac{(-1)^n}{n}\right)$$
The first question to ask is whether the terms converge to zero. If not, the series cannot possibly converge.