Chapter 11
Following a Sequence, Winning the Series
In This Chapter
Knowing a variety of notations for sequences
Telling whether a sequence is convergent or divergent
Expressing series in both sigma notation and expanded notation
Testing a series for convergence or divergence
Just when you think the semester is winding down, your Calculus II professor introduces a new topic: infinite series.
When you get right down to it, series aren’t really all that difficult. After all, a series is just a bunch of numbers added together. Sure, it happens that this bunch is infinite, but addition is just about the easiest math on the planet.
But then again, the last month of the semester is crunch time. You’re already anticipating final exams and looking forward to a break from studying. By the time you discover that the prof isn’t fooling and really does expect you to know this material, infinite series can lead you down an infinite spiral of despair: Why this? Why now? Why me?
Don’t worry. In this chapter, I show you the basics of series. First, you wade into these new waters slowly by examining infinite sequences. When you understand sequences, series make a whole lot more sense. Then I introduce you to infinite series. I discuss how to express a series in both expanded notation and sigma notation, and then I make sure you’re comfortable with sigma notation. I also show you how every series is related to two sequences.
Next, I introduce you to the all-important topic of convergence and divergence. This concept looms large, so I give you the basics in this chapter and save the more complex information for Chapter 12. Finally, I introduce you to a few important types of series.
Introducing Infinite Sequences
A sequence of numbers is simply a bunch of numbers in a particular order. For example:
1, 4, 9, 16, 25, ...
π, 2π, 3π, 4π, ...
2, 3, 5, 7, 11, 13, ...
2, –2, 2, –2, ...
0, 1, –1, 2 –2, 3, ...
When a sequence goes on forever, it’s an infinite sequence. Calculus — which focuses on all things infinite — concerns itself predominantly with infinite sequences.
Each number in a sequence is called a term of that sequence. So in the sequence 1, 4, 9, 16, ... , the first term is 1, the second term is 4, and so forth.
Understanding sequences is an important first step toward understanding series, so read on to get started.
Understanding notations for sequences
The simplest notation for defining a sequence is a variable with the subscript n surrounded by braces. For example:
You can reference a specific term in the sequence by using the subscript:
The notation {an} with braces refers to the entire sequence.
The notation an without braces refers to the nth term of the sequence.
When defining a sequence, instead of listing the first few terms, you can state a rule based on n. (This is similar to how a function is typically defined.) For example:
Sometimes, for increased clarity, the notation includes the first few terms plus a rule for finding the nth term of the sequence. For example:
This notation can be made more concise by appending starting and ending values for n:
This last example points out the fact that the initial value of n doesn’t have to be 1, which gives you greater flexibility to define a number series by using a rule.
Looking at converging and diverging sequences
Every infinite sequence is either convergent or divergent. Here’s what each means:
A convergent sequence has a limit — that is, it approaches a real number.
A divergent sequence doesn’t have a limit.
For example, here’s a convergent sequence:
This sequence approaches 0, so:
Thus, this sequence converges to 0.
Here’s another convergent sequence:
This time, the sequence approaches 8 from above and below, so:
In many cases, however, a sequence diverges — that is, it fails to approach any real number. Divergence can happen in two ways. The most obvious type of divergence occurs when a sequence explodes to infinity or negative infinity — that is, it gets farther and farther away from 0 with every term. Here are a few examples:
–1, –2, –3, –4, –5, –6, –7, ...
ln 1, ln 2, ln 3, ln 4, ln 5, ...
2, 3, 5, 7, 11, 13, 17, ...
In each of these cases, the sequence approaches either ∞ or –∞, so the limit of the sequence does not exist (DNE). Therefore, the sequence is divergent.
A second type of divergence occurs when a sequence oscillates between two or more values. For example:
0, 7, 0, 7, 0, 7, 0, 7, ...
1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, ...
In these cases, the sequence bounces around indefinitely, never settling in on a value. Again, the limit of the sequence does not exist, so the sequence is divergent.
Introducing Infinite Series
In contrast to an infinite sequence (which is an endless list of numbers), an infinite series is an endless sum of numbers. You can change any infinite sequence to an infinite series simply by changing the commas to plus signs. For example:
The two principal notations for series are sigma notation and expanded notation. Sigma notation provides an explicit rule for generating the series (see Chapter 2 for the basics of sigma notation). Expanded notation gives enough of the first few terms of a series so that the pattern generating the series becomes clear.
For example, here are three series defined using both forms of notation:
As you can see, a series can start at any integer.
As with sequences (see “Introducing Infinite Sequences” earlier in this chapter), every series is either convergent or divergent:
A convergent series evaluates to a real number.
A divergent series doesn’t evaluate to a real number.
To show how evaluation of a series connects with convergence and divergence, I give you a few examples. To start out, consider this convergent series:
Notice that as you add this series from left to right, term by term, the running total is a sequence that approaches 2:
This sequence is called the sequence of partial sums for this series. I discuss sequences of partial sums in greater detail later in “Connecting a Series with Its Two Related Sequences.”
Thus, this series converges to 2.
Often, however, a series diverges — that is, it doesn’t equal any real number. As with sequences, divergence can happen in two ways. The most obvious type of divergence occurs when a series explodes to infinity or negative infinity. For example:
This time, watch what happens as you add the series term by term:
–1, –3, –6, –10, ...
Clearly, this sequence of partial sums diverges to negative infinity, so the series is divergent as well.
A second type of divergence occurs when a series alternates between positive and negative values in such a way that the series never approaches a value. For example:
So here’s the related sequence of partial sums:
1, 0, 1, 0, ...
In this case, the sequence of partial sums alternates forever between 1 and 0, so it’s divergent; therefore, the series is also divergent. This type of series is called, not surprisingly, an alternating series. I discuss alternating series in greater depth in Chapter 12.
Later in this chapter, I show you how to decide whether certain important types of series are convergent or divergent. Chapter 12 also gives you a ton of handy tools for answering this question more generally. For now, just keep this important idea of convergence and divergence in mind.
Getting Comfy with Sigma Notation
Sigma notation is a compact and handy way to represent series.
Okay — that’s the official version of the story. What’s also true is that sigma notation can be unclear and intimidating — especially when the professor starts scrawling it all over the blackboard at warp speed while explaining some complex proof. Lots of students get left in the chalk dust (or dry-erase marker fumes).
At the same time, sigma notation is useful and important because it provides a concise way to express series and mathematically manipulate them.
In this section, I give you a bunch of handy tips for working with sigma notation. Some of the uses for these tips become clearer as you continue to study series later in this chapter and in Chapters 12 and 13. For now, just add these tools to your toolbox and use them as needed.
Writing sigma notation in expanded form
As it stands, you may not have much insight into what this series looks like, so expand it out:
As you spend a bit of time generating this series, it begins to grow less frightening. For one thing, you may notice that in a race between the numerator and denominator, eventually the numerator catches up and pulls ahead. Because the terms eventually grow greater than 1, the series explodes to infinity, so it diverges.
Seeing more than one way to use sigma notation
Virtually any series expressed in sigma notation can be rewritten in a slightly altered form. For example:
You can express this series in sigma notation as follows:
Alternatively, you can express the same series in any of the following ways:
Discovering the Constant Multiple Rule for series
In Chapter 4, you discover that the Constant Multiple Rule for integration allows you to simplify an integral by factoring out a constant. This option is also available when you’re working with series. Here’s the rule:
Σ can = c Σ an
For example:
To see why this rule works, first expand the series so you can see what you’re working with:
Working with the expanded form, you can factor a 7 from each term:
Now express the contents of the parentheses in sigma notation:
As if by magic, this procedure demonstrates that the two sigma expressions are equal. But, this magic is really nothing more exotic than your old friend from grade school, the distributive property.
Examining the Sum Rule for series
Here’s another handy tool for your growing toolbox of sigma tricks. This rule mirrors the Sum Rule for integration (see Chapter 4), which allows you to split a sum inside an integral into the sum of two separate integrals. Similarly, you can break a sum inside a series into the sum of two separate series:
Σ (an + bn) = Σ an + Σ bn
For example:
A little algebra allows you to split this fraction into two terms:
Now the rule allows you to split this result into two series:
This sum of two series is equivalent to the series that you started with. As with the Sum Rule for integration, expressing a series as a sum of two simpler series tends to make problem-solving easier. Generally speaking, as you proceed onward with series, any trick you can find to simplify a difficult series is a good thing.
Connecting a Series with Its Two Related Sequences
Every series has two related sequences. The distinction between a sequence and a series is as follows:
A sequence is a list of numbers separated by commas (for example: 1, 2, 3, ...).
A series is a sum of numbers separated by plus signs (for example: 1 + 2 + 3 + ...).
When you see how a series and its two related sequences are distinct but also related, you gain a clearer understanding of how series work.
A series and its defining sequence
The first sequence related to a series is simply the sequence that defines the series in the first place. For example, here are three series written in both sigma notation and expanded notation, each paired with its defining sequence:
When a sequence {an} is already defined, you can use the notation Σ an to refer to the related series starting at n = 1. For example, when
Understanding the distinction between a series and the sequence that defines it is important for two reasons. First, and most basic, you don’t want to get the concepts of sequences and series confused. But second, the sequence that defines a series can provide important information about the series. See Chapter 12 to find out about the nth-term test, which provides a connection between a series and its defining sequence.
A series and its sequences of partial sums
You can learn a lot about a series by finding the partial sums of its first few terms. For example, here’s a series that you’ve seen before:
And here are the first four partial sums of this series:
You can turn the partial sums for this series into a sequence as follows:
In general, every series Σ an has a related sequence of partial sums {Sn}. For example, here are a few such pairings:
This rule should come as no big surprise. After all, a sequence of partial sums simply gives you a running total of where a series is going. Still, this rule can be helpful. For example, suppose that you want to know whether the following sequence is convergent or divergent:
What the heck is this sequence, anyway? Upon deeper examination, however, you discover that it’s the sequence of partial sums for a very simple series:
This series, called the harmonic series, is divergent, so you can conclude that its sequence of partial sums also diverges.
Recognizing Geometric Series and P-Series
At first glance, many series look strange and unfamiliar. But a few big categories of series belong in the Hall of Fame. When you know how to identify these types of series, you have a big head start on discovering whether they’re convergent or divergent. In some cases, you can also find out the exact value of a convergent series without spending all eternity adding numbers.
In this section, I show you how to recognize and work with two common types of series: geometric series and p-series.
Getting geometric series
A geometric series is any series of the following form:
Here are a few examples of geometric series:
In the first series, a = 1 and r = 2. In the second, a = 1 and . And in the third, a = 3 and .
If you’re unsure whether a series is geometric, you can test it as follows:
1. Let a equal the first term of the series.
2. Let r equal the second term divided by the first term.
3. Check to see whether the series fits the form a + ar2 + ar3 + ar4 + ....
For example, suppose that you want to find out whether the following series is geometric:
Use the procedure I outline as follows:
1. Let a equal the first term of the series:
2. Let r equal the second term divided by the first term:
3. Check to see whether the series fits the form a + ar2 + ar3 + ar4 + ... :
As you can see, this series is geometric. To find the limit of a geometric series a + ar + ar2 + ar3 + ..., use the following formula:
So the limit of the series in the previous example is:
When the limit of a series exists, as in this example, the series is called convergent. So you say that this series converges to .
In some cases, however, the limit of a geometric series does not exist (DNE). In that case, the series is divergent.
An example makes clear why this is so. Look at the following geometric series:
In this case, a = 1 and r = . Because r > 1, each term in the series is greater than the term that precedes it, so the series grows at an ever-accelerating rate.
This series illustrates a simple but important rule of thumb for deciding whether a series is convergent or divergent: A series can be convergent only when its related sequence converges to zero. I discuss this important idea (called the nth-term test) further in Chapter 12.
Similarly, look at this example:
This time, a = 1 and r =. Because r < –1, the odd terms grow increasingly positive and the even terms grow increasingly negative. So the related sequence of partial sums alternates wildly from the positive to the negative, with each term further from zero than the preceding term.
A series in which alternating terms are positive and negative is called an alternating series. I discuss alternating series in greater detail in Chapter 12.
For example, suppose that you’re asked to calculate the value of this series:
The fact that you’re being asked to calculate the value of the series should tip you off that it’s geometric. Use the procedure I outline earlier to find a and r:
So here’s how to express the series in sigma notation as a geometric series in terms of a and r:
At this point, you can use the formula for calculating the value of this series:
Pinpointing p-series
Another important type of series is called the p-series. A p-series is any series in the following form:
Here’s a common example of a p-series, when p = 2:
Here are a few other examples of p-series:
A geometric series has the variable n in the exponent — for example, .
A p-series has the variable in the base — for example .
As with geometric series, a simple rule exists for determining whether a p-series is convergent or divergent.
I give you a proof of this theorem in Chapter 12. In this section, I show you why a few important examples of p-series are either convergent or divergent.
Harmonizing with the harmonic series
When p = 1, the p-series takes the following form:
This p-series is important enough to have its own name: the harmonic series. The harmonic series is divergent.
Testing p-series when p = 2, p = 3, and p = 4
Here are the p-series when p equals the first few counting numbers greater than 1:
Because p > 1, these series are all convergent.
Testing p-series when p =
When , the p-series looks like this:
Because p ≤ 1, this series diverges. To see why it diverges, notice that when n is a square number, say n = k2, the nth term equals . So this p-series includes every term in the harmonic series plus many more terms. Because the harmonic series is divergent, this series is also divergent.
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