Our goal in this part of the course is to approximate complicated functions with much simpler ones, namely polynomials. We have seen one example where the function could be expressed in terms of its Taylor series
for values . To interpret this statement, however, we need to understand when the sum of an infinite number of numbers is finite (in this case, we say the series converges). That is the goal of this section. Basically, we will aim to present some simple techniques for determining when a series will converge.
Geometric SeriesWe have already seen that a geometric series will converge precisely when the ratio of successive terms is smaller than one in absolute value:
converges when . For instance, the series converges.
The harmonic seriesThe harmonic series is obtained by summing the reciprocals of the positive integers:
You might think that this series converges because the addends are approaching zero. However, the issue of convergence is a little more subtle than that. We can study this series by comparing it to an integral.
In this diagram is shown the graph ; the area under this graph is shaded in orange. Also shown are rectangles built from the function. From the picture, it is clear that the area of the rectangles is greater than the area under the curve.
How much area is represented by the rectangles? The leftmost rectangle has height 1 and base 1 so its area is 1. The next rectangle has height and base 1 so its area is . You can imagine that the n th rectangle will have height and base 1 giving an area of . Putting this all together, we see that the area of the rectangles is ; in other words, the area of the rectangles represents the harmonic series. Since this area is larger than the area under the graph, we have:
In other words, the area of the rectangles is infinite and so the harmonic series diverges.
More generally, we can compare series of the form to the integral in precisely the same way. This leads to the conclusion that
For example, the series
converges. Notice, however, that comparison does not give us a value to which the sum converges.
Alternating seriesWe can modify the harmonic series to produce the alternative harmonic series :
Notice that this looks a lot like the harmonic series only the sign alternates from term to term. In the demonstration below, we will study this example:
Whereas the harmonic series does not converge, the alternating harmonic series does converge due to the alternation of signs. In some sense, two consecutive terms nearly cancel one another out and this results in the series converging.
More generally, suppose that we have a series such that
- and have opposite signs; that is, the signs of the individual terms alternate.
- ; that is, the terms are decreasing in absolute value.
- ; that is, the terms are approaching zero.
Then, in the same way as the alternating harmonic series, we can be sure that the series converges.
FactorialsGiven an integer n, we define n factorial to be the product of the first n integers and we write . By convention, we will take . Notice that
As you can see, factorials grow remarkably fast. In this example, we will consider the series
We can see that this series converges by noticing:
if . In other words, the series we are interested in eventually becomes smaller than a geometric series which converges.
This is interesting because, in addition to showing that the series converges, it shows that it converges to a number smaller than 3. The following demonstration studies will show you the sum of the first few terms in this series:
Even more remarkably, this demonstration suggests that the series converges to the number . We will soon see why this is. In the meantime, you may want to practice this idea by showing that the series
converges no matter what value of is used.