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## 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?