help? power-series?

Question

Answer 1

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 .

Find

By the ratio test, the series is convergent when .

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

Step 2:

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.

Comments

Step 3:

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.

The interval of convergence is .

Solution:

The interval of convergence is .

Answer 2

2)

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 .

Find .

By the ratio test, the series is convergent when .

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

Step 2:

Substitute in the series .

Consider  .

Find .

Doesnot exist.

The series is divergent by the test of divergence.

Thus, the series is divergent at by the test of divergence.

Comments

Continued....

Step 3:

Substitute in the series .

The series is divergent by the test of divergence.

Thus, the series is divergent at by the test of divergence.

The interval of convergence is .

Solution:

The interval of convergence is .