Find the area covered by the three grilled cheese sandwiches shown below.
Each sandwich has width 1. The areas are
a1(1) = a1
a2(1) = a2
a3(1) = a3
The total area covered by the three sandwiches is
a1 + a2 + a3.
If we add the areas of those three grilled cheese sandwiches, we get the value of the finite series
a1 + a2 + a3.
The graph of those sandwiches is a visualization of the series:
We can use the same idea to visualize any series. Think of each term an of the series as the area of a grilled cheese with width 1 and height an. Then the sum of all terms of the series is the sum of the areas of all the sandwiches.
So far we've been looking at series with positive terms because they're easy to draw and it makes sense to think about their area. If we think about weighted area instead, we can apply this idea to any series of numbers, positive and/or negative. In this case, we could consider grilled cheese sandwiches made using American cheese as positive and those made using Swiss cheese negative. When we add them together, we can get a positive or negative area, just like integrals.
Visualize the harmonic series
on a 2-D graph. Draw the rectangle for each term an on the interval [n – 1, n], so that we get a right-hand sum instead of a left-hand sum.
The rectangle for a1 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 nth 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.
Draw 2-D graphs to visualize the series
using (a) a left-hand sum and (b) a right-hand sum.
(a) Draw the rectangle for the term an = n2 on the interval [n, n + 1]:
(b) Draw the rectangle for the term an = n2 on the interval [n – 1, n]: