Analysis: Absolute convergence, rearranging terms, power series question

[pcvt]

Homework Statement

I can't figure out what my professor means in the last two lines:

I'm trying to prove why the product of two analytic functions is analytic, and I think that I am going to need to use a similar construction.

The Attempt at a Solution

So far to prove the product is analytic, I have said that for
f(x) convergent for x<R1
g(x) convergent for x<R2
Then, (f*g)(x) is convergent for x<min(R1,R2), then, I think I can say that for some rearrnagment of terms, (f*g)(x) = sum n=0 to infinity of (c_n x^n),

but I'm not sure if or why I could make that last step.

Thanks!

 

[mighty2000]

It seems like you are on the right track with your proof. The last two lines of your professor's statement may be referring to the fact that for the product of two analytic functions to be analytic, both functions must be convergent in their respective domains. This means that the power series representation of the product must also be convergent. Therefore, by rearranging the terms of the power series, you can show that the product is also in the form of a power series, which is a characteristic of analytic functions.

I hope this helps clarify the last two lines for you. Keep up the good work!