8.3: Series of Real Numbers

Infinite Series

Preview Activity $8.3.1$ shows that an approximation to $e$ using a linear polynomial is 2, an approximation to $e$ using a quadratic polynomial is $2.5\text{,}$ and an approximation using a cubic polynomial is 2.6667. If we continue this process we can obtain approximations from quartic (degree 4) and quintic (degree 5) polynomials, giving us the following approximations to $e\text{:}$

We see an interesting pattern here. The number $e$ is being approximated by the sum

for increasing values of $n\text{.}$

We can use summation notation as a shorthand¹ for writing the sum in Equation ($8.3.1$) to get

We can calculate this sum using as large an $n$ as we want, and the larger $n$ is the more accurate the approximation ($8.3.2$) is. This suggests that we can write the number $e$ as the infinite sum

This sum is an example of an infinite series. Note that the series ($8.3.3$) is the sum of the terms of the infinite sequence $\left\{\frac{1}{n!}\right\}\text{.}$ In general, we use the following notation and terminology.

We use summation notation to identify a series. If the series adds the entries of a sequence $\{a_n\}_{n \geq 1}\text{,}$ we will write the series as

or

Each of these notations is simply shorthand for the infinite sum $a_1 + a_2 + \cdots + a_n + \cdots\text{.}$

Is it even possible to sum an infinite list of numbers? That doing so is possible in some situations shouldn't come as a surprise. We have already examined the special case of geometric series in the previous section. Moreover, the definite integral is defineed as the limit of a sequence of finite sums, which provides insight into how we'll think of infinite sums generally. As we investigate other infinite sums, there are two key questions we seek to answer: (1) is the sum finite? and (2) if yes, what is its value?

The example in Activity $8.3.2$ illustrates how we define the sum of an infinite series. We construct a new sequence of numbers (called the sequence of partial sums) where the $n$th term in the sequence consists of the sum of the first $n$ terms of the series. If this sequence converges, the corresponding infinite series is said to converge, and we say that we can find the sum of the series. More formally, we have the following definition.

In other words, the $n$th partial sum $S_n$ of a series is the sum of the first $n$ terms in the series,

We investigate the behavior of the series by examining the sequence

of its partial sums. If the sequence of partial sums converges to some finite number, we say that the corresponding series converges. Otherwise, we say the series diverges. From our work in Activity $8.3.2$, the series

appears to converge to some number near 1.54977. We formalize the concept of convergence and divergence of an infinite series in the following definition.

The early terms in a series do not influence whether or not the series converges or diverges. Rather, the convergence or divergence of a series

is determined by what happens to the terms $a_k$ for very large values of $k\text{.}$ To see why, suppose that $m$ is some constant larger than 1. Then

Since $a_1+a_2+ \cdots + a_m$ is a finite number, the series $\sum_{k=1}^{\infty} a_k$ will converge if and only if the series $\sum_{k=m+1}^{\infty} a_k$ converges. Because the starting index of the series doesn't affect whether the series converges or diverges, we will often just write

when we are interested in questions of convergence/divergence as opposed to the exact sum of a series.

In Section 8.2 we encountered the special family of infinite geometric series. Recall that a geometric series has the form $\sum_{k=0}^{\infty} ar^k\text{,}$ where $a$ and $r$ are real numbers (and $r \ne 1$). We found that the $n$th partial sum $S_n$ of a geometric series is given by the convenient formula

and thus a geometric series converges if $|r| \lt 1\text{.}$ Geometric series diverge for all other values of $r\text{.}$

It is generally a difficult question to determine if a given nongeometric series converges or diverges. There are several tests we can use that we will consider in the following sections.

The Divergence Test

The first question we ask about any infinite series is usually “Does the series converge or diverge?” There is a straightforward way to check that certain series diverge, and we explore this test in the next activity.

The result of Activity $8.3.3$ is the following important conditional statement:

If the series $\sum_{k = 1}^{\infty} a_k$ converges, then the sequence $\{a_k\}$ of $k$th terms converges to 0.

It is logically equivalent to say that if the sequence $\{a_k\}$ does not converge to 0, then the series $\sum_{k = 1}^{\infty} a_k$ cannot converge. This statement is called the Divergence Test.

Note well: be very careful with the Divergence Test. This test only tells us what happens to a series if the terms of the corresponding sequence do not converge to 0. If the sequence of the terms of the series does converge to 0, the Divergence Test does not apply: indeed, as we will soon see, a series whose terms go to zero may either converge or diverge.

The Integral Test

The Divergence Test settles the questions of divergence or convergence of series $\sum a_k$ in which $\lim_{k \to \infty} a_k \neq 0\text{.}$ Determining the convergence or divergence of series $\sum a_k$ in which $\lim_{k \to \infty} a_k = 0$ turns out to be more complicated. Often, we have to investigate the sequence of partial sums or apply some other technique.

Next, we consider the harmonic series ²

The first 9 partial sums of this series are shown following.

This information alone doesn't seem to be enough to tell us if the series $\sum_{k=1}^{\infty} \frac{1}{k}$ converges or diverges. The partial sums could eventually level off to some fixed number or continue to grow without bound. Even if we look at larger partial sums, such as $\sum_{n=1}^{1000} \frac{1}{k} \approx 7.485470861\text{,}$ the result isn't particularly convincing one way or another. The Integral Test is one way to determine whether or not the harmonic series converges, and we explore this test further in the next activity.

The ideas from Activity $8.3.5$ hold more generally. Suppose that $f$ is a continuous decreasing function and that $a_k = f(k)$ for each value of $k\text{.}$ Consider the corresponding series $\sum_{k=1}^{\infty} a_k\text{.}$ The partial sum

can always be viewed as a left hand Riemann sum of $f(x)\text{,}$ using rectangles with bases of width 1 and heights given by the values $a_k\text{.}$ A representative picture is shown at left in Figure $8.3.2$. Since $f$ is a decreasing function, we have that

Taking the limit as $n$ goes to infinity shows that

Therefore, if the improper integral $\int_{1}^{\infty} f(x) \ dx$ diverges, so does the series $\sum_{k=1}^{\infty} a_k\text{.}$

What's more, if we look at the right hand Riemann sums of $f$ on $[1,n]$ as shown at right in Figure $8.3.5$, we see that

So if $\int_{1}^{\infty} f(x) \ dx$ converges, then so does $\sum_{k=2}^{\infty} a_k\text{,}$ which also means that the series $\sum_{k=1}^{\infty} a_k$ also converges. Our preceding discussion has demonstrated the truth of the Integral Test.

The Integral Test compares a given infinite series to a natural, corresponding improper integral and says that the infinite series and corresponding improper integral both have the same convergence status. In the next activity, we apply the Integral Test to determine the convergence or divergence of a class of important series.

The Limit Comparison Test

The Integral Test allows us to determine the convergence of an entire family of series: the $p$-series. However, we have seen that it is often difficult to integrate functions, so the Integral Test is not one that we can use all of the time. In fact, even for a relatively simple series such as $\sum \frac{k^2+1}{k^4+2k+2}\text{,}$ the Integral Test is not an option. In what follows we develop a test that applies to series of rational functions by comparing their behavior to the behavior of $p$-series.

Activity $8.3.7$ illustrates how we can compare one series with positive terms to another whose convergence status we know. Suppose we have two series $\sum a_k$ and $\sum b_k$ with positive terms and we know the convergence status of the series $\sum a_k\text{.}$ Recall that the convergence or divergence of a series depends only on the terms of the series for large values of $k\text{,}$ so if we know that $a_k$ and $b_k$ are proportional for large $k\text{,}$ then the two series $\sum a_k$ and $\sum b_k$ should behave the same way. In other words, if there is a positive finite constant $c$ such that

then $b_k \approx ca_k$ for large values of $k\text{.}$ So

Since multiplying by a nonzero constant does not affect the convergence or divergence of a series, it follows that the series $\sum a_k$ and $\sum b_k$ either both converge or both diverge. The formal statement of this fact is called the Limit Comparison Test.

The Limit Comparison Test shows that if we have a series $\sum \frac{p(k)}{q(k)}$ of rational functions where $p(k)$ is a polynomial of degree $m$ and $q(k)$ a polynomial of degree $l\text{,}$ then the series $\sum \frac{p(k)}{q(k)}$ will behave like the series $\sum \frac{k^m}{k^l}\text{.}$ So this test allows us to determine the convergence or divergence of series whose terms are rational functions.

The Ratio Test

The Limit Comparison Test works well if we can find a series with known behavior to compare. But such series are not always easy to find. In this section we will examine a test that allows us to examine the behavior of a series by comparing it to a geometric series, without knowing in advance which geometric series we need.

We can generalize the argument in Activity $8.3.9$ in the following way. Consider the series $\sum a_k\text{.}$ If

for large values of $k\text{,}$ then $a_{k+1} \approx ra_k$ for large $k$ and the series $\sum a_k$ is approximately the geometric series $\sum ar^k$ for large $k\text{.}$ Since the geometric series with ratio $r$ converges only for $-1 \lt r \lt 1\text{,}$ we see that the series $\sum a_k$ will converge if

for a value of $r$ such that $|r| \lt 1\text{.}$ This result is known as the Ratio Test.

Note well: The Ratio Test looks at the limit of the ratio of consecutive terms of a given series; in so doing, the test is asking, “is this series approximately geometric?” If so, the test uses the limit of the ratio of consecutive terms to determine if the given series converges.

We have now encountered several tests for determining convergence or divergence of series.

• The Divergence Test can be used to show that a series diverges, but never to prove that a series converges.
• We used the Integral Test to determine the convergence status of an entire class of series, the $p$-series.
• The Limit Comparison Test works well for series that involve rational functions and which can therefore by compared to $p$-series.
• Finally, the Ratio Test allows us to compare our series to a geometric series; it is particularly useful for series that involve $n$th powers and factorials.
• Two other tests, the Direct Comparison Test and the Root Test, are discussed in the exercises.

One of the challenges of determining whether a series converges or diverges is finding which test answers that question.

Summary

• An infinite series is a sum of the elements in an infinite sequence. In other words, an infinite series is a sum of the form
  $$a_1 + a_2 + \cdots + a_n + \cdots = \sum_{k=1}^{\infty} a_k$$

  where $a_k$ is a real number for each positive integer $k\text{.}$

• The $n$th partial sum $S_n$ of the series $\sum_{k=1}^{\infty} a_k$ is the sum of the first $n$ terms of the series. That is,
  $$S_n = a_1+a_2+ \cdots + a_n = \sum_{k=1}^{n} a_k\text{.}$$
• The sequence $\{S_n\}$ of partial sums of a series $\sum_{k=1}^{\infty} a_k$ tells us about the convergence or divergence of the series. In particular
  • The series $\sum_{k=1}^{\infty} a_k$ converges if the sequence $\{S_n\}$ of partial sums converges. In this case we say that the series is the limit of the sequence of partial sums and write
    $$\sum_{k=1}^{\infty} a_k =\lim_{n \to \infty} S_n\text{.}$$
  • The series $\sum_{k=1}^{\infty} a_k$ diverges if the sequence $\{S_n\}$ of partial sums diverges.