absolute convergence- help?

Question

Answer 1

(1)

Step 1:

The series is .

Use the comparison test to determine series is absolutely convergent, conditionally convergent or divergent.

The Comparison Test :

Suppose that and  are series with positive terms.

(i) If  is convergent and  for all n , then  is also convergent.

(ii) If is divergent and for all n, then is also divergent.

Step 2:

The series is .

The range of the cosine function is .

So .

-series test :

The -series is convergent if and divergent if .

So from the -series, is convergent.

So from the comparison test is convergent.

is absolutely convergent since .

Solution :

is absolutely convergent.

Answer 2

(2)

Step 1:

The series is .

Use the ratio test to determine series is absolutely convergent, conditionally convergent or divergent.

Ratio Test :

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

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

(iii) If , then the ratio test is inconclusive.

Step 2:

The series is .

Consider .

Apply ratio test :

Since , the series is conditionally convergent.

Solution :

is conditionally convergent.

Answer 3

(3)

Step 1:

The series is .

Use the ratio test to determine series is absolutely convergent, conditionally convergent or divergent.

Ratio Test :

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

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

(iii) If , then the ratio test is inconclusive.

Step 2:

The series is .

Consider .

Apply ratio test :

Since , the series is conditionally convergent.

Solution :

is conditionally convergent.