Use mathematical induction to verify that the following integral converges for any positive integer n

Basis (n=1)

We will use integration by parts

In order to calculate the above integral we shall use L'Hospital's rule:

 First we rewrite the limit so we could use L'hospital's rule.

Now we differentiate.

Let us now return to calculating the integral.

As we can...

Basis (n=1)

We will use integration by parts

In order to calculate the above integral we shall use L'Hospital's rule:

 First we rewrite the limit so we could use L'hospital's rule.

Now we differentiate.

Let us now return to calculating the integral.

As we can see the integral converges to 1.

Let us assume that integral  converges for all 

Step (n=k+1)

We will once again use integration by parts.

From the assumption we know that the above integral converges, therefore we only need to show that  also converges. 

If we now apply L'Hospital's rule  times, we will get

Thus, we have shown that the integral converges for  concluding the induction.

QED  

The image below shows graphs of the function under integral for different values of  We can see that -axis is asymptote for all of the graphs meaning that the function converges to zero for all  The only difference is that the convergence gets a little bit slower as  increases and so the area under the graph increases as well. However, the area remains finite for all  as we have already concluded.