Question

Use the Comparison Theorem to determine whether the integral is convergent or divergent.

Answer 1

Step 1:

The integral is .

Rewrite the integral as

Consider the second integral on right side .

Now we can apply comparison value theorem for above integral.

Consider the fact and it implies that  .

Step 2:

Comparison theorem:

Suppose that are continuous functions with ,

1. If is convergent, then is convergent.

2.If is divergent, then is also divergent.

Here and

Hence is a finite value, it is convergent.

By comparison theorem, is convergent.

is convergent, it follows that is also convergent.

and are convergent, then is convergent.

Solution:

is convergent.