Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence...

Integral test is applicable if is positive and decreasing function on interval where  

If is convergent then the series is also convergent.

If is divergent then the series is also divergent.

For the  series , we have then we may let the function:

.

 The graph of f(x) is:

 As shown on the graph, is positive on the interval .  Based on the behavior of the graph as x increases, the function eventually decreases. We can confirm this by applying First Derivative test.  To determine the derivative of the function, we may apply the Product rule for differentiation: .

Let: then

         then

Note:  

                     

                     

Applying the Product rule for differentiation, we get:

         

         

Solve for critical values of by applying .

 Apply zero-factor property:

then

Using test point after , we get:

.

When , then the function is decreasing for the given integral.

Then from the interval . Since the function is ultimately decreasing on the interval we may apply the integral test:

To determine the indefinite integral of , we may apply u-substitution by letting:  or then or .

The integral becomes:

                     

                     

Apply the integration formula for exponential functions:

                     

Plug-in on  , we get:

                     

Applying definite integral formula:

                             

                             

Applying  , we get:

                              

                             

                             

                            or

The   implies that the integral converges.

Conclusion:

The integral  is convergent therefore the series must also be convergent.