Chapter 11 - Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises - Page 557: 67

The series $\sum_{n=1}^{\infty}\frac{1+(-1)^n}{n } $ diverges.

We have the series: $\sum_{n=1}^{\infty}\frac{1+(-1)^n}{n } $ Note that when $n$ is odd, then the series becomes $\frac{1-1}{n}=0$. However, if $n$ is even, then we get $\frac{1+1}{n}=\frac{2}{n}$, which is a divergent harmonic series. Thus, our series diverges as well.