Problem 62E

Exponential function In Section 9.3, we show that the power series for the exponential function centered at 0 is

Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series.

f(x) = x2ex

Solution 62E

Step 1:

We are given that for

We have to find the power series for

Therefore

It is a fact that the x’s outside the series is not going to affect the interval of convergence.

The radius of convergence in the resulting series is

Therefore the interval of convergence is

The power series representation is