Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.g(x) = In (1 ? 3x) using

Solution 37EStep 1:In this problem we need to find the the power series representation for centered at 0 by differentiating or integrating the power series for We already know a power series for ,namely for Therefore power series representation foris Step 2:Now let us find the interval of convergence of Use Ratio test to determine the interval of convergence.If there exists an N so that for all , and and1. If L<1, converges.2. If L>1, diverges.3. If L=1, the test is inconclusive.We have