Convergence of Power Series with Coefficients Multiplied by n

[ohreally1234]

Homework Statement

Suppose sum(a_n*x^n) represents a power series with radius of convergence (-R, R). Is it true that the series sum(n*a_n*x^n) is convergent? Prove or give a counter example.

Homework Equations

The Attempt at a Solution

Let b_n = n*a_n*x^n
Using ratio test:
lim[b_(n+1)/b_n] = lim [(n+1)/n] * [lim x*a_(n+1)/a_n] < 1 because lim x*a_(n+1)/a_n <1 from hypothesis.

Any gaps in logic?

 

[Dick]

I think that's correct.