Combining power series Use the power series representation

to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.

g(x) = x3 ln(l − x)

Solution 28EStep 1:

Given Using the power series representation for , we get

Step 2:

Series isand

To determine the radius of convergence we use ratio test which states:

The ratio test states that:a. If then the series convergesb. If then the series divergesc. If or the limit does not exist then the test is inconclusive.

Calculating L, we get For the series to be convergent .

So, the radius of convergence is 1.