Divergence of the following series.

I have this series

$$\sum_{n=1}^{\infty} \frac{2^n}{n(1-2^n)}$$

Well since it's a series on non-positive terms I decided to analyse the series of non-negative terms:

$$\sum_{n=1}^{\infty} \frac{2^n}{n(-1+2^n)}$$

Well what I basically did was to compare it to the series

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

using the limit test. I reached to a limit equal to 1 and so the series have the same nature.

Since one of the series is the harmonic series (divergent) the series we're analysing is divergent.

But my question now is:

Since $\sum_{n=1}^{\infty} \frac{2^n}{n(-1+2^n)}$ is divergent can I conclude $\sum_{n=1}^{\infty} \frac{2^n}{n(1-2^n)}$ is divergent? Because I don't find any property that says that (only if the series is convergent).

Can someone help me?

Also do you know if there is any easier test I can apply studying this series?

Thanks!

• The two series are equal modulo a multiplicative constant (-1) so if one is divergent, so is the other. The "equivalent" test is already very easy, isn't it ? – Evariste Jun 4 '16 at 12:23

Saying that $\sum_{n=1}^N s_n$ diverges means that it has no limit when $N\to\infty$. If $\sum_{n=1}^N s_n$ has no limit, neither has $-\sum_{n=1}^N s_n=\sum_{n=1}^N (-s_n)$.