I shall do,

This is both a type I and II improper integral. It is a type II because it is not-defined at a point.

We can subdivide the region into as follows,

If both integral exists, then the integral above exists as well.

Let us do the second summand,

.

We shall find the anti-derivative (or at least try) for,

Let,

and .

Thus,

and .

Thus,

When we take the limit we have,

The first summand certainly exists, (I hope you see why).

The second summand can be "squeezed",

Thus, it is suffienct (for convergence) for,

Those, two integral converge (p-series for p=2>1).

Thus,

exists.

Now the question remains to show,

Expand this in the Taylor series,

Integrate, term by term,

The substituting and then taking will result in 0. And substituting 1 will result in an infinite sum,

Which will certainly converge, because the terms are alternating, decreasing, and their limit is zero. Leibnize alternating test gaurenttes convergence.