To prove the convergence of U in a direct comparison test, we must find a series that we know converges AND that we can prove is greater than U. The thinking here is that if we can prove that something bigger than U is not infinity, then U is definitely not infinity.
Similarly, if we want to prove that U diverges, we can look for a series which we know diverges and is smaller than U. If something smaller than U is infinity, then U must definitely be infinity as well.
Instead of comparing the series with another series, we can also compare it with an improper integral. The “improper” part of “improper integral” means that the integral is a definite integral with either upper or lower limit of integration (or both) being plus or minus infinity.
Since we want integrals to compare to infinite series, we'll generally consider improper integrals of the form
So what will we integrate in order to prove that our series converges or diverges? We can't stick just any function in for f(x) in the integral above.
Let's recall that the area between the curve f(x) and the x-axis between x = a and x = b is equal to the definite integral of f(x) between x = a and x = b.
It is evident since the curve is always above or at the height of the rectangles that the area of the curve must be greater than that of the rectangles. Therefore if the area under the curve is finite, i.e., if the definite integral of the curve between 1 and ∞, then the total area of the rectangles must also be finite.
Perhaps it is slightly more of a leap to accept the proposition that if the area under the curve is infinite, so must be H, the value of the series.
The area under the curve can only be a bit (finitely) more than that of the rectangles, since at every integer value of x the curve is equal to the height of the rectangle, and we can work out a formula for the area between the curve and the top of the rectangle.
This is the intuition behind the integral comparison test for convergence of an infinite series. Remember to convert the series index, i in the examples above, to a continuous variable x and integrate with respect to x, setting the limits of integration according to the values i can take in the series. If the result is finite, then the series converges, but, as should be evident from the example above, the series is not equal to the integral.
Good luck on the AP Calculus Exam!