1/x

------Diverges.

1/X^2

------Converges.

How quickly the convergent function approaches the x axis is important, generally, if you wish to take a finite area under the curve. Though both functions never touch the x axis the convergent function comes as close as is possible quickly and it's area under the curve is measurable. The divergent function has an infinite area as it approaches the x axis too slowly to have the area under the curve be measurable.