Part VII: Infinite Series

Sometimes we must work with sums that have an infinite number of terms; such a  sum is referred to as an infinite series. This might happen, for example, when we are computing the exact area of a non-rectilinear region (e.g, a circle). As far back as in the times of ancient Greece, mathematicians were finding the area of circles by inscribing and circumscribing polygons. They found that, if a square was inscribed in a circle and another square circumscribed about the circle, the area of the circle had to be between the areas of the two squares. As the number of sides of the inscribed and circumscribed polygons were increased, the approximations for the area of the circle became more and more exact. However, no matter how many sides the polygons had, the approximation would never be exact. The process for obtaining the exact area from the approximations is what gave rise to the study of sums that had an infinite number of terms. These types of sums require tools from mathematical analysis in order for us to understand and work with them.