This applet illustrates the integral test for convergence of infinite series. Initially a sequence where is a positive, continuous, non-increasing function of is shown. The functions can be changed but make sure it is positive, continuous and non-increasing. A rectangle for each is also shown. The sum of the areas of the rectangles is equal to and the start location of the rectangles can be shifted with the start slider. The number of terms can also be changed with the N slider.
Advancing to the next step shows .
With start = 0 and neglecting the area of , how does the integral compare with the sum of the rectangle areas? With start = 1, how does the integral compare with the sum?
Advancing to the next step shows an equation representing the integral and sum relations and one more step shows the numerical values of the sums and integral.

By adding to the left two terms the inequality can be rewritten as . Then if , the series is also bounded from the right side inequality. Also, if , then from the left part of the inequality, the series is also unbounded. Or in other words, if the integral converges ( is bounded ) then the sum converges and if the integral diverges ( is unbounded ) the sum diverges.
Does the sum need to start at 1 for the integral test to be valid?
What would be required if for the same conclusions to be true?