Comparison with Formulas Examples

Let f (x) = e^-x – 5. Does

converge or diverge?

It looks like this integral probably converges, since f (x) looks a lot like e^-x, and we know

converges.

We have to be careful though. It's true that

e^-x – 5 < e^-x,

but e^-x – 5 isn't bounded below by 0, so the integral could diverge to -∞.

In fact that's exactly what happens. We have to make sure that our function is also bounded below by something that converges. Otherwise the integral could be equal to -∞ as was the case here.

Let . Does

converge or diverge?

Since  and

diverges, so does

Does

converge or diverge?

We know that

-1 < sin x < 1.

This means

The part we care about is the part that says

Since

diverges, the integral

diverges also.