$\sum_{n=1}^{\infty} \frac{(1/2) + (-1)^{n}}{n}$ converges or diverges? How to check if the series $$\sum_{n=1}^{\infty} \frac{(1/2) + (-1)^{n}}{n}$$ converges or diverges?
When $n$ is odd, series is $\sum \frac{-1}{2n}$
When $n$ is even, series is $\sum \frac{3}{2n}$
This series is similar to the series 
$$\sum \frac{-1}{2(2n-1)} + \frac{3}{2(2n)}$$
$$= \sum \frac{8n-6}{8n(2n-1)}$$
Which is clearly divergent.
So, the given series is divergent.
Is this method right?
Please, suggest if there is some easier way.

The idea is correct, but not correctly expressed. Asserting that the given series converges is equivalent to the assertion that the sequence$$\left(\sum_{n=1}^N\frac{\frac12+(-1)^n}n\right)_{N\in\mathbb N}$$converges. If it does, then the sequence$$\left(\sum_{n=1}^{2N}\frac{\frac12+(-1)^n}n\right)_{N\in\mathbb N}$$converges too. But it follows from your computations that it doesn't.
