19. What is the interval of convergence of the power series

Let,

So that,

Thus, by the Ratio Test the given series converges if |(2x - 4)| < 1, that is, converges for (3/2) < x < (5/2). However, the Ratio Test does not tell us the behavior of the series at the endpoints. Therefore, substituting the endpoints into the given series yields the following series:

The left end-point results in the harmonic series, which is divergent, and the right end-point results in a series that converges, that can be shown by the Alternating Series Test.

Hence, the interval of convergence of the given power series is,