I am unsure how to approach this question: 
A series of the form $\sum\limits_{n=1}^\infty c_n$ is known to converge. If each $c_n$ may have values that are positive or negative, which one of the following statments must also **always$$ be true:
I) $\sum\limits_{n=1}^\infty (-1)^nc_n$ must converge
II) $\sum\limits_{n=1}^\infty \frac{c_n}{2^n}$ must converge
III) $\sum\limits_{n=1}^\infty 2^nc_n$ must converge
I have tried using the limit comparison test, but I get an answer of 0, so I cannot yield anything from it. How do I approach determining if these series containing the series cn are also convergent or not?
Thanks for your help!
I. Is not necessarily true. Consider $c_n=\frac{(-1)^{n}}{n}$, then
$$ \begin{align} \sum_{n=1}^{\infty}(-1)^nc_n &= \sum_{n=1}^{\infty}(-1)^n\frac{(-1)^{n}}{n}\\ &= \sum_{n=1}^{\infty}\frac{(-1)^{2n}}{n}\\ &= \sum_{n=1}^{\infty}\frac{1}{n} \end{align} $$
which diverges.
II. Is true.
III. Is not necessarily true. Take $c_n = \frac{1}{2^{n^2}}$, then
$$ \sum_{n=1}^{\infty}2^n\frac{1}{2^{n^2}} = \sum_{n=1}^{\infty}\frac{1}{2^n} $$
which converges.