Find the area covered by the three grilled cheese sandwiches shown below.

Answer

Each sandwich has width 1. The areas are

*a*_{1}(1) = *a*_{1}

*a*_{2}(1) = *a*_{2}

*a*_{3}(1) = *a*_{3}

The total area covered by the three sandwiches is

*a*_{1} + *a*_{2} + *a*_{3}.

If we add the areas of those three grilled cheese sandwiches, we get the value of the finite series

*a*_{1} + *a*_{2} + *a*_{3}.

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 *a*_{n} of the series as the area of a grilled cheese with width 1 and height *a*_{n}. 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.