Does $\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$ converge or diverge?

Question

I'm having trouble figuring out if the following series converges or diverges.

$$\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$$

Here's my thinking:

$$\frac{2}{n} \leq \frac{3+(-1)^n}{n}$$

Since $\sum_{n=1}^{\infty} \frac{2}{n}$ diverges, then so does $\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$

Is that correct?

Answer 1

Yes your prove is perfectly fine, indeed note, as an alternative

$$\sum_{n=1}^{N} \frac{3+(-1)^n}{n}=\sum_{n=1}^{N} \frac{3}{n}+\sum_{n=1}^{N} \frac{(-1)^n}{n}$$

and taking the limit $N\to \infty$ the first series on the RHS diverges whereas the second one converges (by Leibniz).

Answer 2

Note that $$\frac{2}{n}\leq\frac{3+(-1)^n}{n},$$ so by the comparison criteria, your series diverges.