Is this method of finding range of x for which given series is convergent, wrong?

Question

There was this question in our midsem question paper:

We had to find out the range of values of x (x is positive) for which the given series is convergent,

Given series was $\sum_{n = 1}^{\infty} (a + b + c)$ where a, b and c are some functions in terms of x and n.

Now, b was something like this: $\cfrac{1}{(3n-1)^{(x+1)}}$.

Now this series is convergent for all positive real values of x.

Then, I mentioned that:

1. For the series (a+b+c) to be convergent and b proved to be convergent, series (a+c) must also be convergent (since, sum of a convergent and divergent series cannot be convergent).

2. Then, I applied power series formula for radius of convergence and obtained the answer.

Now, my answer and the correct answer match, but this question which is worth 18 credit points, my instructor has given me zero... What I would like to know is, whether there I any mistake in my method or not.

Thanks a lot

Vishwesh

PS: Sorry for not mentioning the problem earlier, I had to search the question paper. Here is the question:

$\sum_{n=1}^{\infty}[\cfrac{x^{3n-2}}{(3n-2)!} + \cfrac{1}{(3n-1)^{x+1}} + x^{3n}]$

Here is my solution (sorry but I will have to upload it as an image):

Answer 1

I suspect that the devilish detail is hidden in the fact that after getting rid of $b$ you still have two summands $a$ and $c$. If you computed the radius of convergence $R_a$ for $a$ and the radius of convergence $R_c$ for $c$, then it follows that $a_n+c_n$ converges within radius $x<\min\{R_a,R_b\}$, and we have divergence for $\min\{R_a,R_c\}<x<\max\{R_a,R_c\}$. However, nothing can be said so straightfirwardly about $\max\{R_a,R_c\}<x$ (not to mention the nedd of special investicgation at $x=R_a$ and $x=R_c$). Since your argument "1" is correct and if you sufficiently reasoned that $b$ is convergent, I suppose that your answer might be worth a strictly positive number of credits. However, depending on the specific problem, this may be only a rather small part of the complete solution. (As mentioned in the comments, it is not really possible to give any conclusive statement without knowing the exact problem statement, your solution, and course requirements)