February 13, Option 9 Overview. Mind Map

Transcription

1 Option 9 Overview Mind Map

2 Return tests - will discuss Wed..1.1 J.1: #1def,2,3,6,7 (Sequences) 1. Develop and understand basic ideas about sequences. J.2: #1,3,4,6 (Monotonic convergence) A quick review: A sequence is a list of number (terms) in a definite order. A sequence is infinite if it contains and infinite number of terms (ie. n Z + ) A sequence is finite if it contains and finite number of terms (ie. n Z +, n < N) {an} represents the sequence of values generated by some rule a, for example, an = 3n + 1 We can often connect a continuous function f(x) to a discrete sequence {an} if f(n) = an. Sequences may or may not have limits! Formally: Limit of a Sequence A sequence {an} has a limit, L, if for every ε > 0 there exists a positive integer N such that an L < ε for all terms an with n > N. In such cases, If the limit exists, we say the sequence converges. Otherwise the sequence diverges. We will state without proof that if a limit exists, then it is unique. Let's develop some theorems for limits of sequences: We begin with the Archimedean Property: For any two line segments of lengths a and b with a < b, it is possible to lay the length a end to end along the length b a finite number of times to create a length equal to or longer than b. Algebraically, For all ε > 0, N Z + such that Nε > 1 Which leads to many other theorems: Proof: The proofs here are the way we prove that a limit exists from the formal definition. The proof is similar to the above. We looked at the integral of the continuous version of two cases last week. Recall that converges for p > 0 and diverges otherwise For a sequence {c n } with c R, n Z + We can also "squeeze" sequences: Squeeze Theorem for Sequences Some other vocabulary: Bounded Sequences A sequence {an} is bounded if and only if an M > 0 such that a n M for all n Z + With this definition, all convergent sequences are bounded. Does it follow that bounded sequences converge? No! Consider sin(n) Some of the properties of limits that we know can be extended to limits of sequences: Finally, we can use properties of functions to help understand limits of sequences if we can relate the sequence to a corresponding function. Let's try some of these ideas

3 Develop and understand monotonic sequences. A few more definitions: 1. A sequence is decreasing if u n u n+1 for all n 2. A sequence is increasing if u n u n+1 for all n 3. A sequence is monotonic if it is increasing or decreasing 4. A sequence is strictly increasing or decreasing if the inequalities above are strict. 5. The number M is a lower bound of the sequence u n if u n M for all n. 6. The number N is an upper bound of the sequence u n if u n N for all n. 7. A sequence is bounded if and only if it has both an upper bound and a lower bound. These definitions result in an important idea: A monotonic sequence converges if and only if it is bounded. So... if you can show that a sequence is monotonic and bounded, then you know it converges. Consider It's worth looking at the first few terms: Is it really decreasing? Since n 1, thus the sequence is decreasing Since it is bounded by the first two terms, the minimum upper bound is 2 Since it is monotonic and bounded, the sequence converges. What does it converge to? How do we show this rather than just state a hunch? The last factor converges to zero. The others are all positive and < 2 so the product is zero. J.1: #1def,2,3,6,7 (Sequences) J.2: #1,3,4,6 (Monotonic convergence)

4 J.1: #1def,2,3,6,7 (Sequences) #6,7 J.2: #1,3,4,6 (Monotonic convergence) #3,4,6.2.1 K.1: #2,4ef,7cd,8,10,12 (Convergence Tests) K.2: #1,2,4 (Error analysis) 1. Understand and use properties of infinite series. 2. Understand and use the comparison test for series. 3. Understand and use the limit comparison test for series 4. Understand and use the integral test for series. 5. Review, understand and use p-series. A series is a sum of a sequence. We will be considering infinite series so assume an infinite sum unless otherwise stated. We have seen arithmetic and geometric series in the core topics. A quick review: Properties of Arithmetic Series 1. Each term differs from the previous by a constant difference d. 2. The n th term is given by un = u1 + d(n - 1) 3. The sum of n terms is given by: Infinite arithmetic series tend to be boring as they diverge Properties of Geometric Series 1. Each term differs from the previous by a constant ratio r. 2. The n th term is given by un = u1r n-1 3. The sum of n terms is given by: Can you develop this formula? Under some conditions, infinite geometric series converge. Sum of an Infinite Geometric Series The limiting sum as n of an infinite series with r < 1 is given by: If r 1, an infinite geometric series diverges. To explore more complicated infinite series, we begin by considering partial sums in which we sum a finite number of terms. If we define the n th partial sum as: Sn = u1 + u un Using the partial sums, the infinite series can be described as: If S exists and is finite, we say the series converges, if S as n, the series diverges. The remaining parts of this section explore how we can determine the convergence or divergence of a series. Necessary condition for convergence It should be self evident that in order for a series to converge, the expression for the general term must approach zero as n gets large. That is we require that: For a series to converge, it must be true that Always check this first! It can save you a lot of time! However, this is not sufficient to determine that the series converges. Consider the harmonic series. Clearly the individual terms approach zero. However, we can rewrite the harmonic series as: Compare this with a thoughtfully constructed series that is similar: Notice that each group of terms in the comparison series is less than the corresponding group in the harmonic series. Furthermore, each group of terms in the comparison series adds to ½. This wise choice makes it easy to see that the comparison series diverges since it's partial sums can be written as 1 + ½k for k > 2. As k gets large, this clearly diverges. Since the comparison series diverges and the corresponding terms in the harmonic series are larger than the comparison series, the harmonic series must also diverge.

5 .2.1 (cont) 1. Understand and use properties of infinite series. 2. Understand and use the comparison test for series. 3. Understand and use the limit comparison test for series 4. Understand and use the integral test for series. 5. Review, understand and use p-series. The strategy we just used to show divergence of the harmonic series is an example of the comparison test which is conceptually similar to the integral comparison test and the squeeze theorem. Precisely stated it is: The Comparison Test (Positive terms only!) This is sometimes called the direct comparison test. But there are times when it does not apply. Consider, for example, We know that is less than the corresponding geometric series But which is geometric with r <1 and hence converges. Thus converges. But what about We suspect it converges but it is not dominated by so we cannot use this test. For this we can prove an alternative comparison test called the Limit Comparison Test. The Limit Comparison Test (Positive terms only!) Why? Since the ratio has a limit L it must be true that for a large enough N, where k and m are positive integers with k < L < m. Thus an is "squeezed" between k and m or kbn < an < mbn. Since all values of an obey this above a certain N, Now, if converges, so does and must converge from the RHS And if diverges, so does and must diverge from the LHS Try to prove parts 2 & 3. Referring back to the relationship between functions and series, we can imagine that if the integral of a decreasing function from some finite value to converges, then so too will the corresponding series. This is a consequence of the exact integral being bounded by the upper and lower Riemann sums. More formally: The Integral Test Proof: Note the limitation that f has to be decreasing and positive for all x greater than 1. The lower limit is 1 to include all terms of the sequence un in the series sum. The proof leads to another useful result: If f(x) is a continuous, positive, decreasing function on [ 1, [, then The book also includes an example proving the convergence properties of the p-series using the integral test. We've seen that series before - you now see how helpful it can be! One more look before we move on...

6 .2 1. Understand and use approximation of the limit of an infinite series by truncation. We've looked at some techniques for deciding whether an infinite series converges, but if it does, what is the value to which it converges? Unlike the case of geometric series, there are many series with no obvious closed form for the limit of the sum. In these cases, we can approximate the limiting value by looking at the partial sum for a large value of n. You will recall that we developed the idea of integrals as areas under curves by considering upper & lower Riemann sums. Just as a Riemann sum is an approximation of an integral, an integral can be thought of as an approximation of a Riemann sum. In particular, if we choose the partition of a Riemann sum to be of width one, we can use integrals to approximate discrete series. Approximating Series with Integrals Given a continuous function on the domain x a, where a is an integer, the corresponding series can be approximated by We can use this idea to estimate the error in approximating an infinite series by a finite partial sum. Consider, for example the series N terms leaving a remainder: which we can approximate with a partial sum of Let's look more closely at the remainder. If (x) is a positive, decreasing function then, from the diagram, we can see that this sum is less than the integral Likewise, if we integrate from N + 1, the sum RN will be larger than the integral Thus we have: Approximation error Example: Estimate the sum of using 10 terms and describe the error. Determine the number of terms needed to obtain an approximation to to within.001% K.1: #2,4ef,7cd,8,10,12 (Convergence Tests) K.2: #1,2,4 (Error analysis)

7 K.1: #2,4ef,7cd,8,10,12 (Convergence Tests) K.2: #1,2,4 (Error analysis).3 1. Understand and use properties of alternating series 2. Understand and use properties of absolute and conditional convergence. 3. Understand and use the ratio test for series convergence Leibnitz tells us that: Alternating Series K.3: #2,4c,6,8,9def,10 (Ratio test) QB #7 We can see this by first looking at the even partial sums, S2N = u1 - u2 + u3 - u4 + u5 - u6...+ u2n-1 - u2n We can group this sum as S2N = (u1 - u2) + (u3 - u4) + (u5 - u6)...+ (u2n-1 - u2n) Now since the terms are decreasing, u1 - u2 > 0, u3 - u4 > 0 and all the terms are positive. We can also show that the sequence is bounded since we can write S2N = u1 - (u2 - u3) - (u4 - u5) - (u6 - u7)...- (u2n-2 - u2n-1) - u2n Again, since the terms decrease, the parentheses are all positive and we have S2N = u1 - (something positive) Having shown that S2N is monotonic and bounded, it must converge to, say, S. Now what about the odd sums, S2N+1. They can be written as S2N + u2n+1. In the limit, this also converges to S since we require that u2n+1 converges to 0. Thus, since both odd and even partial sums converge to the same value, the series converges. Note that in this formulation, the sign of the terms alternates, but since we have written the minus signs in the sum, all the un are positive. This can be confusing. Let's look further at the behavior of alternating series. Notice that the partial sums "jump back and forth" over the convergent value s. From the diagram we can see that the absolute value of the error of any given partial sum from the actual value s is no greater than the next term. Formally that is written as: Error in Alternating Series Example: Determine the convergence or divergence of

8 .3 (cont) 1. Understand and use properties of absolute and conditional convergence We return now to an idea that we saw in our work with improper integrals. That is the idea that if the absolute value of a series converges, then so will the series itself. Formally: Absolute Convergence If both and converge we say that converges absolutely. Clearly, if a series,, converges absolutely, then converges. But it is possible for a series to converge without absolute value converging. Conditional Convergence If converges but does not, we say that converges conditionally. This is helpful because we cannot use the comparison test for alternating series. But we can use it for the absolute value of an alternating series since all terms are then positive. There is a very non-intuitive feature of conditionally convergent series that you should be aware of. Consider, for example, the alternating harmonic series which converges to some value S. If we divide both sides by 2 we get Add the two together and we have But look! This is just a rearrangement of the terms in the original series! So which is it? The answer is both! You cannot rearrange the terms in a conditional infinite series and expect to get the same result! The order of the terms dictates the limit of the partial sums and thus the sum of the series. More generally, certain operations (such as the associative property) that are valid for absolutely convergent series are not valid for conditionally convergent series.

9 .3 (cont) 1. Understand and use the ratio test. With the features of geometric series in mind, consider what we might learn about a series by exploring the ratio of successive terms. In geometric series, this ratio is constant and as long as r < 1 the series converges. If the ratio changes, then we can look at If this limit approaches a value between -1 and 1 (exclusive) then we have a series that approaches a converging geometric series. If not, we approach a diverging geometric series. If the limit of the ratio is equal to 1, we cannot conclude anything. This idea is summarized as D'Alembert's Criterion or the Ratio Test To prove this, we start with the case of L < 1. Since the RHS is a geometric series with 0 < r < 1, it converges. Because it dominates the series and because if converges, then so does (as we will show), by the Comparison Test, our series converges. If L > 1 it must be true that for all n sufficiently large, an + 1 > an. Thus and the series must diverge. Example: All terms are positive so we can ignore the absolute values. Thus the series converges. Try: Thus the series diverges. The ratio test also leads to the useful result: for all k R Can you prove it? K.3: #2,4c,6,8,9def,10 (Ratio test) QB #7,17

10 K.3: #2,4c,6,8,9def,10 (Ratio test) QB # Understand and use power series. K.4: #2-10 even (Power series) QB #8,12,14,18 We now turn to one of the most useful applications of series by examining power series. Power Series A series composed of coefficients multiplied by increasing powers of x is called a power series. Mathematically power series, centered at 0, are represented by We wish to explore the behavior of these series (ie. do they converge or diverge) for various values of x. Let's suppose that we can find a value of x, say x0 for which the series converges. If this is the case, then we know that the general term, anx n 0 as n. Convergence also implies boundedness, so there is some value M for which the absolute value of all the terms in the series are less than M. Now, we can rewrite the series as: and if we look at the series based on the absolute values of these terms we know that it is less than the series But this is M times a geometric series with ratio < 1 so it converges. Thus we know that S', which is less than the one with M in it, converges at x = x0. The previous work shows that if a power series diverges at a value x0 then it will diverge for all values of x > x0. Thus Power Series Convergence If a power series converges for x = x 0, it converges absolutely for all x ]- x 0, x 0 [ Conversely, if the series diverges for x = x 0, it diverges for all x ]- x 0, x 0 [ We can see that the set of points for which a power series converges is symmetric and centered at the origin. If we consider the entire set of points for which the series converges, we can define Radius of Convergence is that value R such that all the points of convergence of the series fall within an interval of convergence x < R The values x = ±R are not part of the interval of convergence and must be dealt with separately. We can find the radius of convergence (and thus the interval of convergence) using the ratio test. To consider the convergence of Consider the related series Since it has all positive terms, we can use the ratio test which says that if then the series converges. For a power series, we require that Example: What about for x = 2? More generally, a power series can be "centered" at some arbitrary place x = a and written in the form: Power Series Centered at "x = a" Summary One last thing: Because series are sums, you can differentiate and integrate series as you would any other sum. This will be helpful next time! A handy summary K.4: #2-10 even (Power series) QB #8,12,14,18

11 K.4: #2-10 even (Power series) QB #8,12, Understand and use Taylor and Maclaurin series. L.1: #1-7 odd (Maclaurin Series) L.2: #2-8 even (Taylor series) QB #19,22,23,24 We now come to an very important application of series - using them more generally to approximate functions. The idea stems from the fact that we can approximate a function in the vicinity of a given value of x by a polynomial which is, itself, a truncated power series. Consider that a polynomial of degree n can be written as Notice that in this formulation the leading coefficient is the last one (an). Now notice that if we want to find the n +1 coefficients of an n th degree polynomial that satisfies a certain condition, we require n +1 pieces of information. We have seen this before when the information given was a requirement for the polynomial to pass through n +1 given points. (Matrix inversion to solve a system of linear equations to yield the coefficients) But now suppose that we require that the value and n successive derivatives of the polynomial at some value x = a match the value and n successive derivatives of a given function at that same value of x = a. This will yield a polynomial that matches the function and also matches it's slope, it's curvature, it's jerk, it's jerk prime (stupid jerk?) and so on. As you might imagine, this polynomial will match the function quite well at values of x that are fairly close to a. In fact, as we increase the degree of the polynomial, it will be a good match to the function farther and farther from a. OK, details! Find a polynomial that fits the graph of ln(x + 1) in the vicinity of x = 0 First, let's look at the successive derivatives of a polynomial In order for the n th derivative of the polynomial to match the n th derivative of the function we need In other words: So back to our function: Thus the polynomial that best fits ln(x + 1) is In the limit as n, we get the result that the function can be represented exactly by the infinite series: In general we have: The Maclaurin Series For a continuous, function infinitely differentiable at x the value of the function is given by: Before going on, there are several Maclaurin series that you should become familiar with: Common Maclaurin Series Notice that we can't do a Maclaurin series for ln(x) since ln(0) is undefined. This is the binomial expansion for rational exponents - a big application. Maclaurin series can give you a very close approximation to a function within only a few terms - as long as you are looking for a value close to x = 0. But what if you want to quickly approximate a function at x = 453, say. Like we did with power series centered at k, we can generalize Maclaurin series in the vicinity of x = a simply by using a horizontal transformation of the variable. The Taylor Series For a continuous, function infinitely differentiable at x the value of the function is given by: Either series will converge to the exact value of f(x) in the limit. But by choosing a convenient nearby value of a you can get a better approximation in fewer terms. Here's what you'll see on the formula sheet.

12 Let's now turn to the accuracy of an estimate. Like we did with general power series, we define a Taylor Polynomial of degree n as the n th partial sum of a Taylor Series. Rn is the remainder after summing n terms. We also call this the error term. The trick here is to find the value of c that gives the maximum error. If x < a, we maximize the value of f (n + 1) (c) for x c a. If x > a, then we find the maximum of f (n + 1) (c) on a c x. This will give an upper bound to the error and is known as the Lagrange Error. Though not mentioned in our book, you can also find the exact error which is given by The proof of this is pretty straightforward but evaluating the error requires that you can do the integral. The error analysis described above can be done on a Maclaurin series by letting c = 0. Let's see an example: Or another common type of question...

13 .2 The question now arises: Does a Taylor series of a function converge to the value of the function in the limit as n approaches? Taylor's Theorem This is only a slight modification what we saw before, with the addition of the idea that the a Taylor series may not actually converge to f(x) but that it will if the remainder (error) converges to zero. One last thought: If two series are added together, the sum has a radius of converges that is the minimum of the radii of convergence of the individual series. This makes sense because both series must converge for the sum to converge, hence the more restrictive radius dominates. 5 L.1: #1-7 odd (Maclaurin Series) L.2: #2-8 even (Taylor series) QB #19,22,23,24

14 L.1: #1-7 odd (Maclaurin Series) L.2: #2-8 even (Taylor series) Discuss #8 QB #19,22,23, Understand and use the binomial series. L.3: #1bc,3 (Binomial Series) L.4: #1-7 odd (Products of series) QB #19,22,23,24 #8 from the HW In summary we found: The Binomial Series which explains our question to QB 8b from the other day! Let's try one: L.1: #1-7 odd (Maclaurin Series) L.2: #2-8 even (Taylor series) QB #19,22,23,24

15 .3 1. Understand and use products of series. We mentioned before that if two series are added the more restrictive radius of convergence of the two series will dictate the radius of convergence of the sum. Not surprisingly the same is true of the product of two series. And the product is a massive FOIL. Formally: Let's try one L.3: #1bc,3 (Binomial Series) L.4: #1-7 odd (Products of series) QB #19,22,23,24 5