Answer
True
Work Step by Step
Given: $\sum c_n6^n$ is convergent.
Using the Ratio Test,
$\lim |\frac{c_{n+1}6^{n+1}}{c_n6^n}|<1$
$\lim |\frac{6c_{n+1}}{c_n}|<1$
$\lim |\frac{2c_{n+1}}{c_n}|<\frac{1}{3}$
It implies that
$\lim |\frac{c_{n+1}(-2)^{n+1}}{c_n(-2)^n}|=\lim |\frac{-2c_{n+1}}{c_n}|=\lim |\frac{2c_{n+1}}{c_n}|<\frac{1}{3}<1$
So, the series $\sum c_n(-2)^n$ is also convergent.
