Does the Series Sum of 1 + (-1)^n/n Converge?

[That Neuron]

Homework Statement

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?".

It listed five series, The answer was this term: 1 + (-1)ⁿ / n.

Homework Equations

\Sigma1 + (-1)ⁿ / n.

The Attempt at a Solution

I find this very confusing simply because whilst separating the series into two separate series, \Sigma1 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 \Sigma1 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!

 

[jbunniii]

\Sigma1 + (-1)ⁿ / n.

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.