The area of the colored rectangles represents the sum of the series

. With the rectangles in the starting (left) position, you can see that

. Slide the rectangles to the right to see that

. Thus the series and the integral converge or diverge together. For a convergent

-series (or any convergent series satisfying the criteria of the integral test), the inequality

holds for all

, where

is the

partial sum. An option in this Demonstration lets you see this fact using

. The gray rectangles represent

, and the colored rectangles represent the "tail",

.