Here are three very similar functions with a definite integral shown. Move points A and B to adjust the definite integrals, see how the areas change. Also note slider n. Adjust slider n to see how each of the integrals behaves as the upper limit goes to the slider value n.
Which functions (if any) converge if the integral was taken from 1 to infinity? Which (if any) diverge? Can you find a pattern? If one (or more) converge, to what value does the improper integral 1 to infinity converge?