Question

Test the series for convergence or divergence.

Answer 1

Step 1 :

Alternating Series Test :

If the alternating series satisfies

(i) 

(ii)

then the series is convergent.

Step 2 :

The series is .

Verify condition (i) :

Since the series is alternating, verify condition (i) and (ii) of the Alternating Series Test.

It is not obvious that the sequence given by is decreasing.

So consider the related function .

Differentiate the function with respect to x .

Since we are considering only positive x , consider .

The above statement is false.

Since the above statement is false, the series does not satisfies condition (i) of alternating series test.

Thus, the given series is divergent.

Solution :

The series is divergent.