Question

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.)

Answer 1

Step 1:

The series is .

Find the interval of convergence using ratio test.

Ratio test :

(i) If , then the series is absolutely convergent.

(ii) If or , then the series is divergent.

Here and .

Step 2:

Find

By the ratio test, the series is convergent when .

Therefore, interval of convergence is  , but need to check convergence at end points also.

Substitute in the series .

.

Above series is a p- series with p =1.

The p- series  is convergent if and divergent if .

Hence the series is divergent at by p- series test.

Step 3:

Substitute in the series .

Above series is a alternating series in which   term is  and for all .

.

Thus, the series is convergent at by alternating series test.

The interval of convergence is .

Solution:

The interval of convergence is .