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 .

Thus, is decreasing on the interval .

This means that, and therefore, , when .

Thus, condition (i) verified.

Step 3 :

As , then .

Evaluate the limits.

.

Thus, condition (ii) is verified.

Thus the given series is convergent by the Alternating Series Test.

Solution :

The series is convergent.