Problem: Is the following True or False:
If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3 \implies:$
a) $\sum_{n=0}^\infty c_n 2^n$ converges.
b) $\sum_{n=0}^\infty c_n 3^n$ converges.
Progress
I found an extra set of notes that had the following theorem, which I could use to answer this question:
If a power series $\sum c_nx^n$ converges for $x=b \neq0$, then it converges for $x$ with $|x|<|b|$.
From the above theorem, we can thus deduce:
a) Since $|2|=2<|-3|=3$, we know $\sum c_n 2^n$ converges. Thus TRUE.
b) Using the same theorem, we can also say that $\sum c_n (-3)^n$ converges $\nRightarrow \sum c_n3^n$ converges, since $|-3| \nless|3|$. Thus (b) is FALSE. To determine whether or not $\sum c_n 3^n$ converges, a separate test must be performed.
Is this reasoning correct?
Your reasoning for (a) is correct, but for (b) it is seriously flawed. "Using the same theorem, we can also say that" ... no, you cannot say anything since the condition of the theorem $|x|<|b|$ is not met. The theorem said "If condition C holds, then conclusion D is true". This does not mean that "if C fails, then D is false"; that would be another theorem to prove.
The correct way to handle question b) is by means of examples, such as $$ \sum \frac{x^n}{3^n n^2} \quad \text{ and } \sum \frac{x^n}{3^n n} $$ in both cases, the series converges for $x=-3$. One of these converges at $3$, the other diverges.