Visualize the harmonic series

on a 2-D graph. Draw the rectangle for each term *a*_{n} on the interval [*n* – 1, *n*], so that we get a right-hand sum instead of a left-hand sum.

Answer

The rectangle for *a*_{1} will still have height 1 and width 1. Instead of putting it on [1,2] we'll put this rectangle on [0, 1]:

The next rectangle, for , will fall on the interval [1, 2]:

and so on.

The area of the *n*th rectangle is still

We draw this rectangle on [*n* – 1, * n*].

If we add up the areas of all the rectangles, we still get

We shifted the rectangles, but still ending up adding the same values.

We've used left- and right-hand sums before to approximate the values of integrals. Now we're using them, along with grilled cheese sandwiches, to visualize series.

Whether we draw the rectangles to the left or the right, the area covered by the rectangles is the sum of the series. It doesn't matter which way we draw the rectangles until we start using the pictures to determine whether a series diverges or converges.