we know the series from n=0 to infinity of c(sub n)*3^n converges

1. TRUE OR FALSE
the series from n=0 to infinity of c(sub n)*2^n must converge

2. TRUE OR FALSE
The series from n=0 to infinity of c(sub n)*4^n must converge

3. TRUE OR FALSE
the series from n=0 to infinity of c(sub n) must converge

MY ANSWERS I WANT CHECKED
1-true
2-false
3-true

To determine the convergence of the series, we can use the ratio test:

1. For the series from n=0 to infinity of c(sub n)*3^n, if the limit as n approaches infinity of (c(sub n+1)/c(sub n))*3/2 is less than 1, then the series converges.

2. For the series from n=0 to infinity of c(sub n)*2^n, we can apply the ratio test by finding the limit as n approaches infinity of (c(sub n+1)/c(sub n))*2/2. If this limit is less than 1, then the series will converge. However, since the multiplier (2/2) is equal to 1, the convergence of the series depends solely on the limit of the ratios of the terms, which cannot be determined based on the given information. Therefore, we cannot determine whether the series converges or not based on the given information. Hence, the answer is "undefined" or "cannot be determined."

3. For the series from n=0 to infinity of c(sub n), this series does not involve any exponential terms, so we cannot directly apply the ratio test. Additionally, without further information about the behavior of the sequence c(sub n), we cannot determine if the series converges or not. Therefore, the answer is "undefined" or "cannot be determined."

Based on this analysis, your answers are as follows:
1. True
2. Undefined or cannot be determined
3. Undefined or cannot be determined