# 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.

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Hint: one piece is convergent while the other diverges. –  1015 Jul 3 '13 at 13:24
So, Is this enough to say this series is divergent? –  mtm Jul 3 '13 at 13:25
Yes: if $\sum a_n$ diverges and $\sum b_n$ converges, then $\sum a_n-b_n$ diverges, as well as $\sum a_n+b_n$. Because if $\sum c_n$ and $\sum a_n$ both converge, $\sum c_n+a_n$ converges. –  1015 Jul 3 '13 at 13:29
@julien, thanks. –  mtm Jul 3 '13 at 13:33

We can estimate. Note that $2^k \ge 2k$, and therefore $$\frac{1}{k}-\frac{1}{2^k} \ge \frac{1}{k}-\frac{1}{2k}=\frac{1}{2k}.$$
It is a familiar fact that $\sum \frac{1}{2k}$ diverges. Thus by Comparison so does our series.