Sequences and Series - PDF Free Download

Sequences and Series What do you think of when you read the title of our next unit? In case your answers are leading us off track, let's review the following IB problems. 1

November 2013 HL 2

November 2008 HL 4

May 2013 HL 6

Some history It's this last type, the infinite series that had puzzled mathematicians for centuries, especially the summation of an infinite series. Why do some infinite series add to a number??? Such as: This can be seen by adding the areas in the "infinitely halved" unit square below. 1 1 8

But sometimes the infinite sum is infinite! Such as: (don't worry we will prove this beyond a reasonable doubt). 9

And sometimes the infinite sum is impossible to pin down! Such as: 1 1 + 1 1 +... = Is it 1? Is it 0? Is it neither? What an exciting time to be a mathematician...or a BC Calculus student! By the end of this unit you will have developed an intuitive sense about infinite series and learn tests for convergence. 10

Misconception An infinite series is not an example of addition. Addition of real numbers is a binary operation, adding numbers two at a time. A finite sum of real numbers always produces a real number because it is a finite number of binary additions. An infinite sum of real numbers is something else entirely...which leads us to the following definition. An infinite series is an expression of the form a 1 + a 2 + a 3 +... + a n +..., or. The numbers a 1, a 2, a 3,... are the terms of the series; a n is the n th term. 11

The partial sums of the series form a sequence S 1 = a 1 S 2 = a 1 + a 2 S 3 = a 1 + a 2 + a 3 S n = of real numbers, each defined as a finite sum. If the sequence of partial sums has a limit L as n, we say that the series converges to the sum L and = L. Otherwise, we say that it diverges. 12

Example 1: Does the series 1 1 + 1 1 + 1 1 +... converge? 13

Example 2: Does the series converge? 14

There is an easy way to identify some divergent series. If the infinite series converges, then This means that if the series must diverge. This is referred to as the n th term test. Does it work? Let's revisit Example 2: 15

Do all infinite geometric series converge? 17

Example 3: Tell whether each series converges or diverges. If it converges, give its sum. (a) (b) (c) (d) 18

How are we feeling about series thus far? Confused Happy Stunned We have just started our journey into infinite series but we know a lot about the convergence and divergence of an entire class of series (geometric series). Let's continue our journey by incorporating x. 19

Representing Functions by Series Let's stick with infinite geometric series for a little bit. What can you conclude about the series below? vs. So what can be said about the following series? 20

Graphic Representation 21

Power Series An expression of the form is a power series centered at x = 0. An expression of the form is a power series centered at x = a. The term center. is the n th term; the number a is the An example of power series, you ask? What about our geometric series? This is a power series centered at x =. It converges on the interval This interval is also centered at x =. This is not a coincidence. This is typical behavior for a power series. A power series will either: 1) converge for all x, 2) converges on a finite interval with the same center as the series, or 3) converges only at the center itself. 22

We have seen that the power series represents the function on the domain ( 1, 1). Can we find power series to represent other functions? Exploration 1: Finding Power Series for Other Functions Given that is represented by the power series on the interval ( 1, 1), 1. express as a power series and find its interval of convergence. 2. express as a power series and find its interval of convergence. 23

3. express as a power series and find its interval of convergence. 4. express as a power series and find its interval of convergence. 24

Finding a Power Series by Differentiation So far we have only represented functions by power series that happen to be geometric. The partial sums that converge to those power series are polynomials and we can apply calculus to polynomials so it would appear that the calculus of polynomials would also apply to power series. Example 4: Given that is represented by the power series on the interval ( 1, 1), find a power series to represent. 25

Theorem: Term by Term Differentiation and Integration If converges for some interval of convergence, then the series obtained by differentiating the series for f(x) term by term f '(x) = converges for the same interval of convergence and represents f '(x) on that interval. If the series for f(x) converges for all x, then so does the series for f '(x). 26

Example 5: Finding a Power Series by Integration 27

Theorem: Term by Term Integration If converges for, then the series obtained by integrating the series for f(x) term by term converges for and represents on that interval. If the series for f(x) converges for all x, then so does the series for the integral. 28

Example 6: Given that find a power series to represent ln(1 + x). 29

Example 7: 30

Identifying a Series So far we have been finding power series to represent functions. Let us try to find the function that a given power series represents. Exploration: Define a function f by a power series as follows: 1. Find f '(x). 2. Find f(0). 3. What well known function do you suppose f(x) is? 4. Graph the first three partial sums. What appears to be the interval of convergence? 5. Graph the next three partial sums. Did you underestimate the interval of convergence? 31

Taylor Series Here we are going to learn a more general technique for constructing a power series. First, let's start by constructing a polynomial. Exploration: Construct a polynomial with the following behavior at x = 0: 32

Exploration 2: Let's see if our general form for the coefficient holds up!!! 33

Example 1: Construct a polynomial that matches the behavior of ln(x + 1) at x = 0 through its first four derivatives. 35

We have just constructed the fourth order Taylor polynomial for the function ln(1 + x) at x = 0. Does it look familiar? 36

We can use this technique to construct Taylor Series about x = 0 for any function, as long as we can keep taking derivatives there. Two functions that are particularly well suited for this are sine and cosine. Example 2: Construct the seventh order Taylor polynomial and the Taylor Series for sin x at x = 0. 37

Now let's graph the first nine partial sums together with y = sin x to see how well we did. 38

Example 3: Construct the sixth order Taylor polynomial and the Taylor series at x = 0 for cos x. 39

Definition: Taylor Series Generated by f at x = 0 (Maclaurin Series) Let f be a function with derivatives of all orders throughout some open interval containing 0. Then the Taylor series generated by f at x = 0 is This series is also called the Maclaurin series generated by f. The partial sum is the Taylor polynomial of order n for f at x = 0. 40

Example 3.5: Find the Maclaurin Series for a previous exploration?. Does it resemble 41

Example 4: Find the fourth order Taylor polynomial that approximates y = cos(2x) near x = 0. 42

Most Useful Maclaurin Series 43

But when are we going to use this??? These polynomial approximations are useful in a variety of ways. First, it is easy to do calculus with polynomials. Second, polynomials are built using only two basic operations (addition and multiplication), so computers can handle them easily. However sometimes being restricted to a power series at x = 0 can be limiting...but why are we limited? We aren't! We can match a power series with f in the same way at any value x = a, provided we can take the derivatives. We get this formula simply by "shifting horizontally". Definition: Taylor Series Generated by f at x = a Let f be a function with derivatives of all orders throughout some open interval containing a. Then the Taylor series generated by f at x = a is The partial sum is the Taylor polynomial of order n for f at x = a. 44

Example 5: Find the Taylor series generated by f(x) = e x at x = 2. 45

Example 6: Find the third order Taylor polynomial for f(x) = 2x 3 3x 2 + 4x 5 (a) at x = 0 (b) at x = 1. 46

How can we find the Power Series of a function from the known Power Series of another function? 1) Find the Maclaurin series for f(x) = cos x by differentiating the Maclaurin series for sin x. 47

2) Using the Maclaurin series for cos x, find the Maclaurin series for. 48

3) Using the Maclaurin series for, find the Maclaurin series for. 49

4) (a) Using the Maclaurin series for find the Maclaurin series for. (b) Find the Maclaurin series for h such that and h(0) = 5. 50

Putting Taylor Polynomials to Use It is nice to know that sin(x) can be found exactly by summing an infinite Taylor series, but if we want to use that information to find sin(3), we will have to evaluate Taylor polynomials until we arrive at an approximation that we are satisfied with. Even our computers and calculators have to deal with this! 51

Example 1: Find a Taylor polynomial that will serve as an adequate substitute for sin(x) on the interval [ π, π]. 52

We want to be able to use Taylor polynomials to approximate functions over the intervals of convergence of the Taylor series and we would like to keep the error of the approximation within specified bounds. Since the error results from truncating the series down to a polynomial (cutting it off after some number of terms), we call it the truncation error. 53

Example 2: Find a formula for the truncation error if we use 1 + x 2 + x 4 + x 6 to approximate the interval ( 1, 1). over 54

How can we handle the error if we were to truncate a nongeometric series? Taylor's Theorem Every truncation splits a Taylor series into two equally significant pieces; 1) the Taylor polynomial P n (x) this gives the approximation 2) and the remainder R n (x) this tells us whether the approximation is any good Taylor's theorem is about both pieces. 55

Taylor's Theorem with Remainder If f has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x in I, where for some c between a and x. Taylor's formula: Remainder of order n: also called the LaGrange form of the remainder 56

Taylor's Theorem is a pretty amazing result. Breaking it down, it says that a function with (n+1) derivatives can be evaluated EXACTLY at a point by a degree (n + 1) polynomial AND tells us exactly how to construct that polynomial. Wow!!! Insert shocked face! Or possibly confused face... 57

Let's do an example to illustrate Taylor's Theorem. Consider expanding e x about x = 0 out to a third degree polynomial. P 3 (x) = Let's use this polynomial to approximate e 2. We get e 2 Taylor's Theorem guarantees the existence of a number c, with 0 < c < 2 such that e 2 = P 3 (x) + R 3 (x) = 58

Let's visualize what is going on. Let's graph f(x) = e x along with P 3 (x). If we use the polynomial to approximate e 2, the error is the length of the short vertical segment from f(x) to P 3 (x). 59

Let's find the length of that vertical segment. Then we can use that to find c. 60

Do we really need to find the value of c? Could we tell anything by just looking at the behavior of R n (x) overall? Example 3: Prove that the series converges to sin(x) for all real x. 61

Example 4: Prove that the series converges to cos(x) for all real x. 62

What are the important consequences of Taylor's Theorem? 1) We saw that we never really care to find the actual value of c (we did it just this once to prove to you that there really will be such a c that exists). Instead we would rather determine some maximum possible value that the magnitude of our remainder term could be for a particular n. Meaning how big could the absolute value of the (n + 1) st derivative of our function be, assuming c is somewhere between a and x? If we can find a bound for the absolute value of that derivative, we can find a bound for the absolute value of the remainder term and a bound on the error. This is called the LaGrange error bound. 63

Remainder Estimation Theorem We were able to use the remainder formula in Taylor's Theorem to verify the convergence of two Taylor series to their generating functions (sin(x) and cos(x)) and we did NOT need an actual value for. Instead we were able to put an upper bound on for all x. which was enough to ensure that Remainder Estimation Theorem If the function f can be differentiated (n+1) times on an interval I containing the number a and if M is an upper bound for on I, meaning for all x in I, then for all x in I. 64

Example 5: The approximation is used when x is small. Use the Remainder Estimation Theorem to get a bound for the maximum error when. 65

What are the important consequences of Taylor's Theorem? 2) If we can determine that the remainder term goes to 0 as n, then we know the Taylor series converges to the function value at the point in question. What's cool is that as long as the (n + 1) st derivative is bounded on the interval in question and independent of n, the remainder term will go to 0 because of GRANNY FEPL! Factorials grow faster than power functions. 66

Consider the following math sentence: Is this always true for all values of x? 77

So far we have had no issue discussing convergence for geometric series and even some of our power series, thanks to Taylor's Theorem. But we need more... 78

The Convergence Theorem for Power Series There are three possibilities for with respect to convergence. 1) There is a positive number R such that the series diverges for but converges for. The series may or may not converge at either of the endpoints x = a R and x = a + R. 2) The series converges for every x. (R = ) 3) The series converges at x = a and diverges elsewhere. (R = 0) R = radius of convergence 79

Test 1: n th Term Test for Divergence The most obvious requirement for convergence of a series is that the n th term must go to zero as n approaches infinity. diverges if fails to exist or is different from zero. Examples: Using the n th Term Test for Divergence, what conclusion can you draw about the convergence or divergence of the given series? 1) 2) 3) 4) 80

Test 2: The Direct Comparison Test Let be a series with no negative terms. (a) converges if there is a convergent series with a n c n for all n > N, for some integer N. (b) diverges if there is a divergent series of nonnegative terms with a n d n for all n > N, for some integer N. 81

Examples: Use the Direct Comparison Test to determine whether the series converges or diverges. 1) 2) 3) 82

Definition: Absolute Convergence If the series of absolute values converges, then converges absolutely. Theorem: If converges, then converges. We use this when the series has some positive and negative terms but it doesn't alternate from positive to negative. More on that later! Example: Show that converges for all x. 83

Examples: 1) 2) 87

Test 3: The Ratio Test Let be a series with. Then, (a) the series converges if L < 1, (b) the series diverges if L > 1, (c) the test is inconclusive if L = 1 and you must perform another test. We also use the Ratio Test to find the Interval of Convergence of a series. 88

Examples: Use the Ratio Test to determine whether the series converges or diverges. 1) 2) 89

Examples: Use the Ratio Test to determine whether the series converges or diverges. 3) 90

Endpoint Convergence The Ratio Test, which is really a test for absolute convergence, establishes the radius of convergence for. But then this is the same as the radius of convergence of So all that remains to be resolved about the convergence of an arbitrary power series is the question of convergence at the endpoints of the convergence when the radius of convergence is a finite, nonzero number. Example: Find the interval of convergence of 91

Example: Find the interval of convergence for each power series. 2) 92

Test 4: The Integral Test Let {a n } be a sequence of positive terms. Suppose that a n = f(n), where f is a continuous, positive, decreasing function of x for all x N (N is a positive integer). Then the series and the integral either both converge or diverge. Note: If the integral exists then it just means the series converges, the value of the integral is not the value of the series. 93

Example: Does converge? 94

The Integral Test can be used to settle the question of convergence for any series of the form:, p is a real constant. We call these series a p series. 95

Exploration: The p Series Test 1) Use the Integral Test to prove that converges if p > 1. 2) Use the Integral Test to prove that diverges if p < 1. 3) Use the Integral Test to prove that diverges if p = 1. 96

The p Series Test If p 1, then the p series diverges. If p > 1, then the p series converges. Note: There is no formula to determine the value to which the series converges. 97

Examples: List the first 3 terms of each series, state the value of p, and state if the series converges or diverges. 1) 2) 3) 98

The Harmonic Series This is the most famous divergent series in mathematics. The p Series Test shows that the harmonic series is just barely divergent; if we increase p to 1.000000001, the series converges! It is difficult to see by just adding up the terms because the partial sums increase at a slower and slower rate. For example, the sum of the first 4 terms edges over 2; 11 terms edges over 3; 227 terms edges over 6; 12,367 terms edges over 10; and about 250 million terms edges over 20. Proof that the harmonic series diverges: Group the terms of the harmonic series as follows: Compare this to the series: or Notice that each term in each grouping of the harmonic series is greater than or equal to each corresponding term in the corresponding grouping of the second series. Since the second series, the harmonic series must also. 99

Example: Find the interval of convergence for the power series. 100

Test 5: The Limit Comparison Test Suppose that a n > 0 and b n > 0 for all n N (N a positive integer) 1) If then and both converge or diverge. 2) If and converges, then converges. 3) If and diverges, then diverges. Example: To show that the series diverges, we can compare this to which is the divergent harmonic series. 101

The Limit Comparison Test works well for comparing "messy" algebraic series to a p series. When choosing an appropriate series to which we want to make a comparison, we disregard all but the highest powers of n in both the numerator and the denominator. Example: 1) Given, choose 2) Given, choose 3) Given, choose 4) Given, choose 102

Example: Determine whether the series converge or diverge, a) b) c) d) 103

What is an Alternating Series? An alternating series is a series whose terms alternate in sign. For example: Alternating Series occur in two ways: 104

Alternating Series Test If a n > 0, then the alternating series and converge, provided both of the following conditions are met: 1) 2) 105

Examples: 1) 2) 3) 106

Illustrating the convergence of the partial sums to their limit L. (Closing in on the sum of a convergent alternating series) 0 S 2 S 4 L S 3 S 1 107

Alternate Series Remainder Theorem Examples: 1) Approximate the sum and error of the series from its first six terms. 108

Example: Prove that the alternating harmonic series is convergent, but not absolutely convergent. Find a bound for the truncation error after 99 terms. 109

Absolute and Conditional Convergence A series is absolutely convergent if A series is conditionally convergent if 110

Examples: 1) 2) 111

Testing for Convergence/Divergence: Which Test to Use The following are guidelines, not rules. 1) Always first see if the terms go to zero. If they don't, then the series diverges (n th term test). If the terms do go to zero, no conclusion can be drawn; try another test. 2) Use the direct comparison test if the series looks like one you know, such as the harmonic series, p series, or geometric series. 3) If you can't use the direct comparison test, but the series looks like one you know, try the limit comparison test. Also, the limit comparison test may be tried if the series is a fraction whose numerator and denominator are polynomials or roots of polynomials. 4) Use the integral test if the series looks like an integral you have done. Don't use the integral test if you can see an easier one to use. 112

Testing for Convergence/Divergence: Which Test to Use 5) Use the ratio test if the series contains combinations of variables in exponents, variables in bases or factorials. 6) Use the root test if the series is an expression to the n th power. 7) If the series has alternating signs (it will have a factor such as ( 1) n or ( 1) n+1 in it), use the alternating series test. 8) If the series contains positive and negative terms (not necessarily alternating), you may also test to see if the series converges absolutely. If it does, then the original series converges. 9) Remember Dominance: n n > Factorials > Exponentials > Powers > Logs x x > x! > b x (b > 1) > x c (c > 0) > ln x Granny FEPL 113

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