help-ratio/root test

Question

Answer 1

1)

Step 1:

The series is .

The Ratio Test:

Let  be a series with nonzero terms.

1) converges absolutely if .

2) diverges if and .

3) The Ratio Test is inconclusive if .

Step 2:

Consider .

Apply Ratio Test.

Comments

Contd..

.

This series is convergent because is less than .

Solution:

The series is convergent.

Answer 2

2)

Step 1:

 The series is .

Consider .

Apply Ratio Test.

.

This series is divergent because is greater than .

Solution:

The series is divergent.

Answer 3

3)

Step 1:

The series is .

Root test :

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

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

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

Consider .

Apply Root Test.

.

Hence, by the root test, the series is convergent.

Solution:

The series is convergent.