# Power Series Convergence

1. Sep 22, 2010

### Brilliant

1. The problem statement, all variables and given/known data
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) $$\sum_{n=0}^{\infty} c_{n}(-4)^n$$

2. Relevant equations
The Power Series: $$\sum_{n=0}^{\infty} c_{n}(x - a)^n$$

3. 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.

2. Sep 23, 2010

### hunt_mat

If you know that:
$$\sum_{n=0}^{\infty} c_{n}4^n$$
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.