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:
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).
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):