5.2 Infinite series

We have seen that a sequence is an ordered set of terms. If you add these terms together, you get a series. In this section we define an infinite series and show how series are related to sequences. We also define what it means for a series to converge or diverge. We introduce one of the most important types of series: the geometric series. We will use geometric series in the next chapter to write certain functions as polynomials with an infinite number of terms. This process is important because it allows us to evaluate, differentiate, and integrate complicated functions by using polynomials that are easier to handle. We also discuss the harmonic series, arguably the most interesting divergent series because it just fails to converge.

Sums and series

An infinite series is a sum of infinitely many terms and is written in the form

But what does this mean? We cannot add an infinite number of terms in the same way we can add a finite number of terms. Instead, the value of an infinite series is defined in terms of the limit of partial sums. A partial sum of an infinite series is a finite sum of the form

To see how we use partial sums to evaluate infinite series, consider the following example. Suppose oil is seeping into a lake such that $1000$ gallons enters the lake the first week. During the second week, an additional $500$ gallons of oil enters the lake. The third week, $250$ more gallons enters the lake. Assume this pattern continues such that each week half as much oil enters the lake as did the previous week. If this continues forever, what can we say about the amount of oil in the lake? Will the amount of oil continue to get arbitrarily large, or is it possible that it approaches some finite amount? To answer this question, we look at the amount of oil in the lake after $k$ weeks. Letting $S_k$ denote the amount of oil in the lake (measured in thousands of gallons) after $k$ weeks, we see that

Looking at this pattern, we see that the amount of oil in the lake (in thousands of gallons) after $k$ weeks is

We are interested in what happens as $k\to \infty .$ Symbolically, the amount of oil in the lake as $k\to \infty$ is given by the infinite series

At the same time, as $k\to \infty ,$ the amount of oil in the lake can be calculated by evaluating $\underset{k\to \infty }{\text{lim}}{S}_k.$ Therefore, the behavior of the infinite series can be determined by looking at the behavior of the sequence of partial sums $\left\{S_k\right\}.$ If the sequence of partial sums $\left\{S_k\right\}$ converges, we say that the infinite series converges, and its sum is given by $\underset{k\to \infty }{\text{lim}}{S}_k.$ If the sequence $\left\{S_k\right\}$ diverges, we say the infinite series diverges. We now turn our attention to determining the limit of this sequence $\left\{S_k\right\}.$

First, simplifying some of these partial sums, we see that