Master AP Calculus AB & BC
Part II. AP CALCULUS AB & BC REVIEW
CHAPTER 11. Sequences and Series (BC Topics Only)
• Introduction to sequences and series, nth term divergence test
• Convergence tests for infinite series
• Power series
• Taylor and Maclaurin series
• Technology: Viewing and calculating sequences and series with a graphing calculator
• Summing it up
As a BC calculus student, you have come a long way since your first limits and derivatives. There may have been a time when you feared the Chain Rule, but that time is long since past. You are now a member of the elite Calculus Club. (First rule of Calculus Club: Don’t talk about Calculus Club.) And your membership is complete with sequences and series.
Once you get an idea of what sequences and series are, we will focus primarily on infinite series. For a couple of sections, you’ll use various tests to determine the convergence of infinite series. Once that is complete, we’ll discuss power series and use Taylor and Maclaurin series to approximate the values of functions. Sound good? Your Jedi training is almost complete ... the Force is strong with this one.
INTRODUCTION TO SEQUENCES AND SERIES, NTH TERM DIVERGENCE TEST
A sequence is basically a list of numbers based on some defining rule. Nearly every calculus book begins with the same example, and it’s so darn fine that I will bow to peer pressure and use it as well. Consider the sequence The number n will take on all integer values beginning at 1, so the resulting sequence of numbers will be
In some cases, for the sake of ease, we will let n begin with 0 instead of 1—but you’ll know exactly when to do that, so don’t get stressed out or confused.
NOTE. The Force is with you always, and it equals mass times acceleration.
Our singular goal in sequences is to determine whether or not they converge. In other words, is the sequence heading in some direction—toward some limiting number? You can graph the sequence above to see that it is headed toward a limit of 0.
NOTE. The Technology section at the end of this chapter explains how to graph sequences and series on your 11-83. You should probably read that before you go any further.
Each term of the sequence is half as large as the term before, and the sequence approaches a limit of 0 very quickly. Mathematically, we write
Because this sequence has a limit as n approaches infinity, the sequence is said to converge; if no limit existed, the sequence would be described as divergent.
Example 1: Determine whether or not is a convergent or divergent sequence.
Solution: The sequence will converge if its limit at infinity exists and will diverge if the limit does not exist.
This is a rational function with equal degrees in the numerator and denominator; therefore, the limit is the ratio of the leading coefficients: Therefore, the sequence converges to 1/2. This is further evidenced by the graph of the sequence below:
A series is similar to a sequence, but in a series, you add all the terms together. Series are written using sigma notation and look like this:
The notation is read “the sum from 1 to infinity of ” and gets its value from the sum of all the terms in the sequence. However, how can you tell if a series with an infinite number of terms has a finite sum? At first glance, it seems impossible—how can you add infinitely many numbers together to get a real sum? Consider the diagram below, which should help you visualize the infinite series :
If the large box represents one unit and you continuously divide the box into halves, the sum of all the pieces will eventually (if you add forever and ever) equal the entire box. Thus,
You won’t be able to draw pictures like this for the majority of series, but, in the next section, you’ll learn a much easier way to find the above sum. Mathematically, you need to know that a series gets its value from the sequence of its partial sums (SOPS). A partial sum is the sum of a piece of the series, rather than the entire thing; it is written Sn, where n is the number of terms being summed. Let’s use good old as an example:
Therefore, the SOPS is .5, .75, .875, .... This is the important thing to remember: If the SOPS converges, then the infinite series converges, and its sum is equal to the limit at which the SOPS converged. For the most part, you will use the fantastic formula alluded to earlier in order to find sums. However, if all else fails, you can use the SOPS to find the sum of a series.
Example 2: Find the sum of the series
NOTE. The series in Example 2 is one example of a telescoping series, the terms in these series cancel out as the SO PS progresses.
Solution: Construct the sequence of partial sums to gain some insight on this series:
See what happened there? The —1/3 and 1/3 will cancel out. In S4, the —1/4 and 1/4 will cancel out. In fact, each partial sum will cancel out another term all the way to infinity, and the only two numbers left will be 1 + 1/2. Therefore, the sum of the series is 3/2. You can use your calculator to calculate S500 to verify that the SOPS is indeed approaching 3/2.
It makes sense that each term in a sequence needs to get smaller if the series is going to converge. You are adding numbers for an infinite amount of time; if you are not eventually adding 0 in this infinite loop, your sum will grow and grow and never approach a limiting value. We showed that the sequence of the terms that make up have a limit of 0 early in this section, and that infinite series has a sum. However, if the limit, as n approaches infinity, of the sequence that forms an infinite series does not equal 0, then that infinite series cannot converge. This is called the nth Term Divergence Test, and it is the easiest way to immediately tell if a series is going to diverge.
nth Term Divergence Test: If then is a divergent series.
Example 3: Show that is divergent.
Solution: Because (and this limit must equal 0 for the series to be convergent), the series diverges by the nth Term Divergence Test. That’s all there is to it. If you think about it, since the limit at infinity is 1, as n approaches infinity, you’d be adding 1 + 1 + 1 + 1 + 1 + 1 forever, and that clearly approaches no limiting or maximum value.
Be careful! Just because , that does not mean that the series will converge! For example, the harmonic series is divergent even though If the limit at infinity is 0, you can conclude nothing from the nth Term Divergence Test. It can only be used to show that a series diverges (if its limit at infinity does not equal 0); it can never be used to show that a series converges.
Directions: Solve each of the following problems. Decide which is the best of the choices given and indicate your responses in the book.
YOU MAY USE YOUR GRAPHING CALCULATOR FOR PROBLEM 4 ONLY.
1. Determine if the sequence converges.
2. Find the nth term of each sequence (in other words, find the pattern evidenced by the sequence), and use it to determine whether or not the sequence converges.
3. Use the nth Term Divergence Test to determine whether or not the following series converge:
4. (a) What is the sum of
(b) Calculate S500 to verify that the SOPS is bounded by the sum you found.
ANSWERS AND EXPLANATIONS
1. The sequence converges if exists. You’ll have to use L’Hopital’s Rule:
The sequence converges.
2. (a) This is the sequence Use L’Hopital’s to show that the sequence converges.
(b) This is the sequence How can we tell that it converges? It is very clear that in fact, this sequence converges to 0 significantly faster than since the former’s denominator will grow larger much faster than the latter’s.
3. (a) since the expression is rational and the degrees of the numerator and denominator are equal. Because the limit does not equal 0, this series diverges by the nth Term Divergence Test.
(b) but this does not necessarily mean that the series is convergent. You will find out that it does converge very soon (in the Integral Test subsection), but you can never conclude that any series converges using the nth Term Divergence Test; it can only be used to prove divergence. Therefore, you can draw no conclusion.
4. (a) This is a telescoping series; if you write out the fourth partial sum (S4), you can see what terms will cancel out in the long run and which ones will remain:
The only terms that will remain as n approaches infinity are so the infinite sum is
(b) Use your calculator to find S500. Of course, this is not the limit at infinity, but it will give us an idea of where SOPS is heading at that point.
S500 is closing in on .8333, but it has all of infinity to get there—so what’s the rush?