Problem 22RE

Power series from the geometric series Use the geometric I series ,for |x|<1 to determine the Maclanrin series and the interval of convergence for the following functions.

Solution 22RE

Step 1:

In this problem we have to use the geometric series for |x|<1, to determine the Maclaurin series of and also we have to find the interval of convergence

We have

Using geometric series for |x| < 1 we get,

Thus the Maclaurin series of is given by

Notice that we replaced both the x in the geometric series with to get the required Maclaurin series and so we will do the same in the interval of convergence.

Thus provided

Thus the interval of convergence for the Maclaurin series is . That is the interval of convergence is