Is Power Series Convergence Related to Other Series Convergence?

[Brilliant]

Homework Statement

If \sum_{n=0}^{\infty} c_{n}4^n is convergent, does it follow that the following series are convergent?

a) \sum_{n=0}^{\infty} c_{n}(-2)^n b) \sum_{n=0}^{\infty} c_{n}(-4)^n

Homework Equations

The Power Series: \sum_{n=0}^{\infty} c_{n}(x - a)^n

The Attempt at a Solution

I was able to work all the problems that asked me to solve for a radius of convergence, but this question seems much different, and I can't think about how to prove or disprove either a or b. Any tips would be much appreciated.

 

[hunt_mat]

If you know that:
<br /> \sum_{n=0}^{\infty} c_{n}4^n<br />
Then apply the ratio test on this to get a relationship between c_{n} and c_{n+1}, then you can use this to check the other series . Look up the alternating series test also.