$\sum_{n=1}^{\infty} \frac{(1/2) + (-1)^{n}}{n}$ converges or diverges?

Question

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.

Answer 1

hint

$$\sum \frac{(-1)^n}{n}$$

is convergent by alternate criteria. $$\frac 12\sum \frac 1n$$ is a divergent Riemann series.

The sum of a convergent and a divergent series is a Divergent one.

Answer 2

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.