How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent? How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent?
I applied integral test and found the series is divergent.  I wonder if there exist easier solutions.

Each partial sum of your series is the difference between the partial sum of the harmonic series, and the partial sum of the geometric series.  The latter are all bounded by 1.  Since the harmonic series diverges, your series does also.
