I have the following alternating series that I would like to determine whether it is absolutely convergent, conditionally convergent, or divergent:
$ \sum\limits^{\infty}_{n=1} \frac{1+2(-1)^n}{n} $
I have applied some tests and I find it reasonable to conclude that it is divergent.
As a sum of two series: $ \sum\limits^{\infty}_{n=1} \frac{1}{n} + \sum\limits^{\infty}_{n=1} \frac{2(-1)^n}{n} $
I believe a convergent series when added to a divergent series, results in a divergent series. If this isn't a fact then I would still be left to say that it is inconclusive.
Using the Alternating Series Test, with: $ a_n = \frac{1+2(-1)^n}{n} $ although this isn't of 'proper form' $ \sum\limits^{\infty}_{n=1} (-1)^n a_n $ the limit of $a_n $ does approach zero as $ n \rightarrow \infty $. As for monotonically decreasing, the limit of the ratio of absolute terms is divergent for $ n $ even and inconclusive for $ n $ odd, which has me concluding divergent by The Ratio Test as well as not monotonically decreasing, where:
$ \lim\limits_{n \rightarrow \infty} \left\lvert \frac{2(-1)^n + 1}{2(-1)^n - 1} \frac{n}{n+1} \right\rvert $
Am I on the right track here? Am I making any really improper assumptions? Was there a better way to go about with the proof?
Thanks!