Main topics for the Final Exam

Transcription

1 Main topics for the Final Exam The final will cover Sections , , , , This is roughly the material from the fourteen homeworks and and six quizzes. In particular, I will expect you to know how to solve each problem from the homeworks and the quizzes. In addition, you will certainly need to know the material from Math Some of this material may feel difficlut to you. If you have gotten before mainly by memorizing formulas, and then plugging into them on the test, be warned that simply will not work this time. It will not always be obvius what the correct formula to use is, and there may be more than one approache that works (although often some approaches will be much less work than others.) If you are having trouble with something, focus on doing problems, not just learning formulas. The only way to really feel this stuff is by doing problems. Your textbook has plenty of examples and exercises - try doing some of them to help you learn the material. Remember, just learning a bunch of theorems doesn t help, unless you also know how to use them (and how to notice when you should use them!!!) Be sure to watch out for the following common mistakes: Remember that the bounds change when you use u-substitution. Also remember that you can only use u substitution f you can find a du term in your original integral - you need to end up with something that is completely in terms of u. Remember that integration is hard. If you think you have found a very easy way to compute integrals, you have probably done something wrong. In particular there is no way to compute dx in general. 1 f(x) If you are trying to use partial fractions, make sure you set up things correctly (one quick check, the number of constants you introduce should equala the degree of the denominator). If you mess this up, you will get stuck. When computing limits, avoid imprecise arguments like n + n 3 n 3, so we can ignore n. Remember, you can usually make these arguments work by simply dividing by (what you suspect is) the largest term. Don t forget that you can divide by the largest term inside radicals as well! When you use L Hopital s rule, make sure to check that you have one of the indeterminate forms 0 0 or. If you need to use L Hopital s more than once, check this at each stage. Remember that 1 is not equal to 1. It is an indeterminate form. How would you calculate that limit. Remember when calculating limits, to take the whole limit at once. In particular, the folloiwng ( ) k is wrong: lim k k = 1 lim k 2 k = 1. Remember that if you take natural logarithms to compute a limit, then your final answer is not the limit you get. You need to take an exponential to get rid of the logarithms. That is, if ln a k π, then a k e π, not π. 1

2 Remember that if a k converges, then a k 0, but not the other way around. If it were that easy to tell if a series converges, most of what we did in the second half of the course would have been pointless. We have several different convergence tests. Make sure you can keep them all straight. When can each of them be used? What do you need to do for each one? What do they tell you? Do they tell that the series converges, or diverges? Do they just tell you that you need to consider a different series (or integral)? Also make sure you know whether you are considering a sequence or a series at each step. For instance, in the root test, are you considering the series a 1/k k of the limit lim? Why does it matter? k a1/k k Remember that the formula you do if the sum starts at k = 1, k = 2, or k = 1000? x k = 1 1 x only works if the sum starts at k = 0. What would Remeber when calculating improper integrals of unbounded functions you want only one sided limits. I.e. if f(x) is unbounded at x = a you want c > a so that you can calculate the proper integral b c f(x)dx. Moreover, in some cases the two sided limit may not even exist, even if the one sided limit ( lim you want, exists. c a +) Make sure you are comfortable with the following: Understand the material from Math In particular: Understand what an integral is, both intuitively (as a sum of infinitely small quantitites), and the actual definition (involving upper and lower sums). Make sure you understand the notations b f(x)dx and f(x)dx. In particular make sure a you understand why the first one is a number, and the second one is a function of x. Pay attention to which one you are asked to compute. If I ask you for a definite integral, your final answer should be a number, If I aks you for an indefinite integral, your final answer should be a function of x (or whatever the dummy variable is.) Know how to use the fundamental theorem of calculus to compute an integral by finding an antiderivative. Know how to use u-substitution to compute integrals. Make sure you are familiar with the forms for definite integrals and indefinite integrals. For definite integtals, remember that bounds change? (This can save you a lot of work by avoiding going back to the original variable.) For indefinite integrals, remember that you final answer should not be a function of u. Know the common trircks for u-substition (letting some complicated expression be u, letting u = x + a, finding a u that will make du appear in the integral, etc.) Don t forget that you also do simple things like u = ax + b, u doesn t have to be a complicated function of x in order to be useful. 2

3 Have a look at section 8.1. Everything in there (except the hyperbolic functions) should seem pretty familiar to you by now. Know how to compute (f 1 ) (x) if f is one-to-one function. Remember that you can sometimes compute this (at least at specific points) without first finding f 1 (x) Know how to define ln x and exp(x), and know their basic properties. Know how to compute f (x) f(x) dx Trig Identities: A number of things we have covered this term involve trig functions, so you will need to know basic trig identities. These will be much more essential this term than they were the last term. Know the simple identities like sin( x) = sin(x), cos x = sin( π 2 x), sin(π x) = sin x, etc. If you understand know the unit circle works, you won t need to memorize all of these. You should be able to come up with them on your own by just lookint at the picture. Know how to find the values of trig functions at certain special values (e.g. those you can get from or right triangles in terms of radians!) Know the Pythagorean identities: sin 2 x + cos 2 x = 1 and 1 + tan 2 x = sec 2 x. They can help you integrare expressions of the form sin n (x) cos m (x) where at least one of n, m is odd.they are also usefull to expressions of the form tan n x sec m x where n is even or m is odd. Know how to take the derivatives of the basic trig functions. In particular, make sure know that (sin x) = cos x, (cos x) = sin x (remember these signs!) and (tan x) = sec 2 x = 1. Also, remember that you can ger the others with the quotient rule if you cos 2 x have forgotten them. Know how to use derivatives of trig functions to help you integrare expressions of the form sin n (x) cos m (x) where at least one of n, m is odd. They are also usefull to expressions of the form tan n x sec m x where n is even or m is odd. Know the double angle identities: sin(2x) = 2 sin x cos x and cos(2x) = 2 cos 2 x 1 = 1 2 sin 2 x and the related half-angle formulas. Focus on how you can use these to integrate expressions of the form sin n x cos m x for n, m both even. It may also help to be somewhat familiar with the sum identities: sin(a+b) = sin a cos b+ cos a sin b, cos(a + b) = cos a cos b sin a sin b and the related product identities for sin a sin b, cos a cos b, sin a cos b. They are helpful in integrating experessions of the form: sin(nx) cos(mx), sin(nx) sin(mx), cos(nx) cos(mx). Inverse Trig Functions: Know how to define the basic inverse trig functions: arcsin and arctan (in particular, what are their domains and ranges?) Know how to define other inverse trig functions (e.g. arccos, arcsec etc.) in terms of these. 3

4 Know how to compute certain values of these functions (if you know some value of a trig function, then you also know a value of the inverse trig function.) Know how to compute things like tan(arcsin x) of arcsin(sin x) (note that this is not always x) etc. If you can t immediately see how to do this with a trig identity, try drawing a picture. Most problems like this can be solved by drawing the init circle or an appropriate rignt triangle. Know how to take the derivatives of inverse trig functions (especially arcsin and arctan) Know how to use this to compute dx and dx a 2 x 2 x 2 +a 2 More generally, make sure you know how to compute things like dx or dx ax 2 +bx+c ax 2 +bx+c by completing the square (keep in mind that how you handle these will depend on whether the polynomial ax 2 + bx + c has real roots or not. Integration by Parts: Understand how the fundamental theorem of calculus and the product rule give the identity u(x)v (x)dx = u(x)v(x) u (x)v(x)dx. a Also know the definite version of this: u(x)v (x)dx = u(x)v(x) a b a u (x)v(x)dx. Remember that these are the limits for x, not for u or v. Know how to use this to sometimes compute f(x)g(x)dx - integrate one factor, and differentiate the other. If you are luck, the new integral will be easier to compute. If it is harder to compute, try the switching the derivative and the function. Remember that this often won t be as simple as memorizing a formula and plugging something in to get your answer. Integration by parts in useless if you can t figure out how to use if effectively. b There will sually be many different ways to break up your function into a product of two functions. Not all of them will be useful. Don t be afraid to try several things of your first guess doesn t work. Remember that there is always at least one way to break up a function as a product: f(x) = (1)f(x). Consider using this if f (x) is a much simple function than f(x), (e.g. f(x) = ln x, arcsin x, arctan x etc.) If using integration by parts doesn t immediately get you the answer, that doesn t mean you should give up on it. Did you end up with an integral which looks simpler? Consider using integration by parts again (e.g.. to compute x 2 e x dx). Maybe the integral you ended up with doesn t look any simpler. Can you find another way to relate if back to the original integral? Think about how we computed e x sin xdx: We used integration by parts twice and ended up with an equation we could solve for e x sin xdx. Don t forger about other techniques (like u-substitution) Is it possible that you could use u-substitution either before or after you use integration by parts? Products and Powers of Trig Functions: b 4

5 Know how to compute the integral sin n x cos m xdx for any m, n, Remember that if either m or n is odd, this is easy, simply use u substitution, with u = sin x, cos x and use the identity sin 2 x + cos 2 x = 1 to write the integral in the form f(u)du for some polynomial f. If m and n are both even, you can use the half-angle identities to decrease the exponents. Know how to hande something like sin(mx) cos(nx)dx (you can use the product identities) Also know how to deal with integrals involving other trig functions e.g. tan 3 xdx, sec 3 xdx. Remember the various tricks you can use, like u-substition, trig identities, and integration by parts. Trig Substitution: Be familiar with the substitutions x = a sin u, x = a tan u, x = a sec u. How do they work? What is dx in each situation? When/why are they useful? (They are useful for simplifying expressions like a 2 x 2, a 2 + x 2, x 2 a 2 respectively.) What interval whould you pick u to lie in? For example, if you let x = a tan u, then you may take u ( π 2, π 2 ) in which case you will always have sec u > 0 and so x 2 + a 2 = a sec u. If you do this to compute an indefinite integral, then you will need to put the answer back in terms of x Make sure you know how to do this, Often simplifying your answer will involve computing things like sin(2 arctan x). Know how to do this (in this case you would want to first use a double angle identity and then a suitable right triangle.) Know what to do with other quadrativ functions. Any quadratic can be written in the form u 2 a 2, u 2 + a 2, a 2 u 2 by completing the square, Make sure you know how to do this and that you can use this to compute an integral involcing some quadratic function. Also keep in mind that an expression like a 2 x 2, a 2 + x 2, x 2 a 2 doesn t mean that you have to use trig substituion, Keep an eye for simpler approachs. You can compute x 4 x dx by setting x = 2 sin u, but is there something much 2 simpler you could do? Integrating Partial Fractions: Remeber that to use the partial fraction decomposition method, you first need a proper rational function. If the degree of the numerator is higher than the degree of the denominator, you will need to first divide the numerator by the denominator beore using the method. Also make sure that quadratics you end up with are really irreducible (for x 2 + bx + c you need b 2 4x < 0) Make absolutely sure that you know how to compute A (x a) dx and Bx+C dx for n (x 2 +bx+c) n x 2 + bx + c. 5

6 If you don t know how to compute these integrals, then you won t be able to use this method. Remember, the first one is easy to compute. Just let u = x a, then you are left with du u Don t try to over complicate this. You don t need to complete the suare, or use n trig substitution of anything like that. This is the sort of things we were computing last term. The second one is harder. Your first step should be to complete the square. Then you are left with u du and 1 du. The first one of these can be done with a (u 2 +a 2 ) n (u 2 +a 2 ) n simple substition (let v = u 2 + a 2 ). The second one requires trig substitution at least of n > 1. In the case of Ax+B dx it will be useful the break the numerator into two parts x 2 +bx+c Ax + B = const.(x 2 + bx + c) + const. Why is this helpful? Partial Fraction Decompostion: Know how to integrate a rational function (write it a sum of partial fractions, then integrate each partial fraction.) Remember that only proper rational fractions can be written as a sum of partial fractions. Know how to write a rational function P (x) r(x) r(x) Q(x) in the form p(x)+ Q(x) where Q(x) is proper. (This is just polynomial long division.) Know how to tell which partial fractions can appear in the decomposition of a given rational function. You can do this by factoring the denominator. A factor of (x a) n corresponds to A factor of (x 2 + bx + c) n corresponds to A 1 (x a) + A 2 (x a) A n (x a) n B 1 x + C 1 (x 2 + bx + c) + B 2x + C 2 (x 2 + bx + c) B nx + C n (x 2 + bx + c) n Remember, the numerator is always constant in the first case (and linear in the second case), regardless of the exponent n. If you have something like Bx+C, then (x a) 2 you have done something wrong. As a check: the number constants you introduce should always equal the degree of the denominator. If you have too many constants, you will not be able to solve for all of them. If you have too few, you will (probably) get a contradiction. If you set up things incorrectly, you will most likely get stuck. Be careful! One you have set things up, you will need to find the values of the constants. Your first step will (probably) be to clear up the denominatos by multiplying by the original denominator. 6

7 There are two main methods for finding the constants: plugging in different values of x (think about whihc ones to use! What whill make your work easier?) or expanding and comparing the coefficients. Know how to use both of these. Also learn how to recognize which one will be less work. Also, be aware that you don t have to restrict yourselfto only using these Sometimes it must be useful to combine them. Maybe you can figure out a few of the coefficients by plugging in values of x, and then find the rest by expanding (before you expand, can you maybe factor out something like (x a) to make your life easier?). Don t be afriad to try other things, if they look like they might make things easier. You can use any method you want, as long as it works. Differential Equations: Know what a differential equation is, and what it means to solve one. Know the difference between a geneal solution (i.e. any function that satisfies the equation) and a particular solution (i.e. one satisfying y(a) = b. Know how to find a particular solution if you know the general solution. Be sure to distinguish between the two types of diffential equations we studied. You won t be able to solve one type by using the method for the other type. Know how to solve a first order linear differential equation: y + p(x)y = q(x). (Multiply by an integrating factor e H(x), to turn the LHS into (e H(x) y). How do you figure out the appropriate H(x) to use? Really check that derivative of e H(x) y is exactly the LHS. If it is not, then you have done something incorectly. Note the plus sign in front of p(x). Know how to solve a separable differential equation (p(x) + q(y)y = 0). Sequences: Know how to tell when your equations is separable, often you will need to rewrite it. Remeber that you need to integrate both sides of an equation with respect to the same variable. Only because dy = y dx one of the integrals turn into integral for y. So pay attention to the integrating variable (differential.) Be careful with things like 1 xdx = ln x + C. Remember that g(x) = x is not differential at x = 0. Be aware that a separable differential equation has as solutions integral curves. They may or may not represent the graph of a function y = y(x). Thing about the curves y 2 + x 2 = C. Know what a sequence is both intuitively and the formal definition as a function. Know how to find the general term of a sequence by noticing the patern of the first few terms or using a recursive formula. Know how to prove your formula using a mathematical induction. 7

8 Know what it means for a sequence to be increasing/decreasing or to be bounded above/below/both. Know how to show that a sequence is increasing by either checking the ratio a n+1 a n or calculating the derivative f (x) for a n = f(n). (In particular, know in which case which method to use). Know the order of increasing of the following expressions log n, n c, c n, n!. The Least Upper Bound Axiom: Know what it means for the number b to be an upper bound (or a least upper bound) for the set S. Know (and understand) the statement of the least upper bound axiom. In particular, remember it applies to real numbers, not sets of rational numbers. Understand why an axiom is different from a theorem - its not something we can prove, its simply a basic property of the real numbers (like x+y = y +x or x(y +z) = xy +xz). It it wasn t true, then we we simply wouldn t be talking about the real numbers anymore. Understand what it is useful forl it gives us a way to show tat some real number exists without being able to explicitely find that number. Sequences: We have been implicitly using it like this last quarter. If we didn t, we wouldn t even be able to talk about basic things like integrals. Know what it means for a sequence to be increasing/decreasing or to be bounded (one sided or both sided) Know how to show that a sequence is increasing by either checking the ratio a n+1 a n or calculating the derivative f (x) for a n = f(n). (In particular, know in which case which method to use). Understand how we can (sometimes) show that a sequence has a limit, without finding an explicit L which satisfies the definition. Understand ow a sequnce can be defined recursively (e.g. a n = 1 + a n 1 ). Know how we can sometimes use induction to determine an explicit formula for a recursive sequence, or to show that a sequence is increasing, or bounded. Know the order of increasing of the following expressions log n, n c, c n, n!. The Least Upper Bound Axiom: Know what it means for the number b to be an upper bound (or a least upper bound) for the set S. Know (and understand) the statement of the least upper bound axiom. In particular, remember it applies to real numbers, not sets of rational numbers. Understand why an axiom is different from a theorem - its not something we can prove, its simply a basic property of the real numbers (like x+y = y +x or x(y +z) = xy +xz). It it wasn t true, then we we simply wouldn t be talking about the real numbers anymore. 8

9 Understand what it is useful forl it gives us a way to show that some real number exists without being able to explicitely find that number. We have been implicitly using it like this last quarter. If we didn t, we wouldn t even be able to talk about basic things like integrals. Know why a bounded increasing (or decreasing) sequence has a limit. Limits of Sequences: Know what is meant by the expression lim n a n = L (both intuitively, and the ε, N- definition. Know the definition of a limit of a sequence and how to use it to prove that a given L is the limit of a given sequence. Know the basic properties of limits: If a n L and b n M, then what is the limit of a n + b n, ca n, a n b n, a n /b n? Know how you can use the pinching theorem to sometimes compute more complicated limits than this. 4n Know how to find the limit of a rational function (e.g. lim 3 5n n 2n 3 7 ). Avoid making imprecise statements like 4n 3 5n 4n 3. Try dividing my the largest power of n, and using the fact that 1/n a 0 for a > 0. Also remember that you can often do this for functions which aren t quite rational (such as ones involving square roots of certain expressions.) You can even do this if your limit involve things like ln n or a n. Remember that ln n is smaller than n a, which is smaller than a n, which is smaller than n!. Explicitely, ln n n a 0 and na a n When trying to compute the limit of f(n) 0 and an n! 0. (for any a > 0.) g(n), identify which term is the largest, and divide the numerator and denominator by that term. That should make both the numerator and denominator have finite limits. If f is continuous, then understand why f( lim ) = lim f(a n) n n If a n is a recursive sequence defined by a n = f(a n 1 ) and if a n has a limit, L, know how to find L. (Hint: If a n L, then lim a n 1 = lim a n = l and f(a n 1 ) L) If a n is a sequence with positive terms, know how to find the limit, by looking at lim n ln a n: If ln a n L, what is lim n a n? If ln a n +, what is lim n a n? If ln a n, what is lim n a n? Consider doing this when it looks like ln a n will be simpler that a n (such as when a n involves a lot of exponents.) Make sure you know how to compute limits (as n ) of the following standard sequences: ( ) n x 1/n, x n, 1 n, ln n a n, a na a, n xn n!, n1/n and 1 + x n 9

10 All of these show up constantly in the material from this term. L Hopital s Rule: Understand why 0 0 is an indeterminate form. That is, if you know f(x), g(x) 0 why can you not conclude anything about lim f(x) g(x)? Understand how L Hopitals rule can help in this situation: If f (x) g (x) γ, then f(x) g(x) γ and understand (intuitively) why this works. Also understand that this works for any choice of γ including γ = ±, and any choice of what x approaches (e.g. x c, x c +, x c, x etc.) Understand why it isn t (technically) correct to state L Hopital s rule as lim f (x) f (x) g (x) lim x c g (x). If lim x c possible for this to still exist? doesn t exist, can we conclude anything about lim x c f(x) x c g(x) = f(x) g(x)? Is it Remember that computing a limit may often invlovle two or more application of L Hopital s rule: if f (x) g (x) is still in the form 0 0, then you can use L Hopital s rule again to evaluate this. Also know L Hopital s rule for. This is technically different theorem, but it looks almost identical, and can be used in basically the same way. Remember that you can ONLY use L Hopital s rule for things in the form 0 0 and! If you try to use L Hopital s rule for anything else, you will get nonsence ( and unless you are paying attention, you won t even notice you ve done anything wrong.) Always be sure to check - never use L Hopital s rule without making sure it works. Other Indeterminate Forms: Know how to recognize, and deal with, indeterminate forms like 0 and You can usually rewrite 0, as 0 0 or f(x) by using f(x)g(x) = 1/g(x) If you have something like, consider re-writing it in some other way, to get 0 0 or. Can you put everything in terms of a common denominator to write it as a fraction? Are logarithms involved so that you can use ln a ln b = ln a b Understand why 0 0, 1, and 0 are all indeterminate forms. How do you deal with these? Consider taking logarithms. Know how to recognize expressions which are not indeterminate forms, such 0 and 0. What is different about these? How would you compute the limit in this case? Improper Integrals: Know what the notations to converge? Know what the notation a f(x)dx and a f(x)dx mean. What does it mean for thes f(x)dx a mean. What does it mean for thes to converge? Remember that we need two different limits to converge here. 10

11 b Also know what What about b a a f(x)dx means when f has a vertical asymptote at b (or at a). f(x)dx when f has asymptotes at both a and b? When it has an asymptote (or asymptotes) in the interior of (a, b)? Remember that for something like this to converge, several different integrals have to converge. If even one of them fails to converge, then the entire integral diverges. Also keep in mind that the fundamental theorem of calculus might not work if f has an asymptote in (a, b). You really need to break these up into multiple integrals in order to compute them. Know how to compute improper integrals. In most cases this isn t much harder than computing ordinary integrals. If you can compute the indefinite integral, then you can simply take the limit. Also remember that you can use u-substitution and integration by parts in exactly the way you would expect. Make sure you understand how this works. When you use u-substitution, what happens to the bounds? Also know that for more complicated integrals it s often convenient to first evaluate the relevant proper defintie integral and only then take the limit. There s no need to keep writing the limit while doing long computations. dx 1 Make sure you know when 1 x p and dx 0 x p converge. It should be p > 1 for one of these and p < 1 for the other, How can you remember which is which, if you happen to forget it? When they do converge, what is their value? dx b What about a x p and dx for a > 0, b > 0. When do they converge? 0 xp Know how you can use the comparison test to tell if an improper integral converges. Find some simpler function (like 1/x p ) to compare it to. This should feel similar to finding the limit of a rational function as x. Remember that ln x < x a for any a > 0 and x sufficiently large. What if you have something like f(x) = 1 you can compare this to 1 x 2 +1 x, but this is larger than f(x) and so it doesn t help. Is there anything you can do to fix this? (i.e. find something smaller than f(x) which diverges?) Also know that you can use the comparison test for situations as a Series: a f(x)dx and g(x)dx where f(x) < g(x) only for x > c > a. What allows you to use the test in such cases? If a n is a sequence, understand what the notation isn t exactly a sum, it s a limit. diverge. a k means. Remember that this Know what it means for this series to converge or 11

12 Know how to compute the sum of some simple series. In particular: A geometric series x k. When does this converge? What are the partial sums? What does the series converge to? 1 A telescoping seties.that is a series like k 1, where most of the terms will k + 1 cancell. Know how to sum these. (Write enough terms in the begining and in the end to see the patern in the cancelations of terms in the n-th partial sum S n, then take the limit.) Don t just focus on memorizing the formulas for these. Make sure you can recognize these when you see them. How can you recognize a geometric series? (Hint: if a n is geometric, what can you say about a n+1 a n?) What can you do if the first term of the series is not 1? What about if it is something of the form 2 k+3 /7 k 2? How can you tell if a series is telescoping? What if it isn t given in the formula a n = f(n) f(n + 1)? Don t forget about partial fractions. These can often help if you are given the sum of a rational function. Understand why 0.a 1 a 2 a 3 = Convergence Tests: Know how to relate the series a k 10 k f(k) to the integral 1 f(x)dx. What conditions does f need to satisfy in order to use the integral test? Remember that this isn t giving us a way to compute the series. The integral converges iff the series converges, but they aren t equal. KNow how to use the integral test to figure out when remember when the p-series converges. Undestand why it doesn t matter where we start the sum i.e. 1 converges. In particular, kp a k will converge iff a k converges. Undestand why it is okay to just write a k without including the k=j bounds. Know that if a k and b k both converge so does (a k + b k ) and ca k Be careful when separating series i.e. (a k + b k ) = a k + b k only if a k and b k both converge. (Why happens if you try separating telescoping series like 1 k 1 k+1 = 1 k 1 k+1? Know how to use the comparison test to figure out if a series converges or diverges (this is almost exactly the same as for improper integrals) 12

13 Know how to use the limit comparison text. This is useful if you know what sequence you want to compare to, but it s difficult to figure out which sequence is larger. Also remember that you need to have a k /b k L where L > 0 in order to use this. If L = 0, what can you conclude? (see HW11) Know how to use the root and ratio tests to check whether the given series converges without using another assistant series. Remember that in either of these tests, if the limit is 1 or doesn t exist, then you can t use that test, and you need to try something esle. Also remember that n 1/n 1, this is very useful when using the root test. Don t just focus on memorizing these tests, think about how you can tell which test to apply to a given series. Does it look easy to integrate the terms? Does it look like it can be compared to a simpler series (like 1 k )? If the terms involve only powers, it may p be simpler to work with a 1/k l. If the terms involve factoriels or factoriels and powers, it may be easier to work with a k+1 a k. Don t forget the Divergence test. It is an easy check whether a series a k converges. Rememer that the converse is not true. What series a k diverges even though lim a k = 0? Absolute and Conditional Convergence: Remember that most of out convergence tests only work for sequence with nonnegative terms. For general series, we need to try something else. Know what it means for a series to absolutely converge or to conditionally converge. Understand why an absolutely convergent series must converge, but a series can converge without absolutely converging. Know how to tell if a series absolutely conveges (Hint: You are now talking about a sequence with nonnegative terms.) Know how to tell if an alternating series converges. Remember that you really do need the sequence to be decreasing. Know how to use partial sums to estimate the sum of alternating series. How close is N ( 1) k a k to ( 1) k a k? Know when to use the Absolute Root or Ratio test and what they tell you about the absolute convergence or divergence of given series Power Series: Know how to find the set of values of x for which a k x k converges (i.e. the interval of convergence.) Understand why this set must always be an interval. Understand why knowing that a k x k converges (or diverges) at some specific x = c tells us that the series converges (or diverges) at other values of x. Understand what the radius of convergence of a power series a k x k is. Why must any power series have one? 13

14 Remember that if the radius of convergence is r, the interval of convergence could be any of ( r, r), [ r, r), ( r, r], or [ r, r]. How do you determine which is the actual interval of convergence? Know how to use the Absolute root or Absolute ratio test to find the radius of convergence of a power series. If f(x) = a k x k, know how to write f (x) and f(x)dx as power series. These all have the same radius of convergence. Do they (necessarily) have the same interval of convergence? Know how to use Abel s Theorem to determine when a power series expansion f(x) = a k x k is valid for x equal to the radius of convergence. Know how to use this to get formulas like = ln 2 or = π 4 Taylor Series: Understand why we would expect the polynomial P n (x) = approximation to the function f(x) for x 0. f (k) (0) x k to be a good k! Understand why just knowing that f(x) P n (x) isn t good enough, we need to know how good an approximation this is. That is, we need to know how big the error term R n (x) (given by f(x) = P n (x) + R n (x)) is. Know the formula for R n (x) from Taylor s theorem involving an integral. Know the simpler formula R n (x) = f (n+1) (c) (n+1)! x n+1 for some c between n and x. (Don t forget this c depends on x and n.) As a tip for remembering this, this looks almost like the next term in the Taylor series. Know how to use this formula to estimate R n (x) (Inequality in the book). If f is a nice function (like e x or sin x) you should be able to show that R n (x) 0 for any x. n f (k) (0) Understand why the fact that f(x) x k f (k) (0) does not imply that f(x) x k. k! k! This only happens when R n (x) 0. It is possible for the series not to converge at some point where f(x) is defined. It is even possible for f(x) to be defied and for the series f (k) (0) x k to converge, but these numbers need not be equal. (see HW13, the last k! problem) Know the basic Taylor series: e x = 1 + x + x2 2! + x3 3! + x4 4! + = cos x = 1 x2 2! + x4 4! x6 6! + = sin x = x x3 3! + x5 5! x7 7! + = 14 x k k! ( 1) k (2k)! x2k ( 1) k (2k + 1)! x2k+1

15 1 1 x = 1 + x + x2 + x 3 + x 4 + = x k ln(1 + x) = x x2 2 + x3 3 x4 4 + = ( 1) k 1 arctan x = x x3 3 + x5 5 x7 7 + = ( 1) k 2k + 1 x2k+1 and know for which values of x they converge. Know how to compute Taylor series in (x a) instead of in x (this should be almost the same as for ordinary Taylor series.) Know how to use Taylor series to compute limits, as an alternative to L Hopital s rule and how to use them to easily compute f (k) (0) for large k. If f(x) = a k x k know why this implies that a k x k is the Taylor series of f(x) This makes it much easier to compute the Taylor series of more complicated functions, lime e x2. Usually if you are asked to compute the Taylor series of some function, you should try doing something like this. If you know the basic Taylor series listed above, you should be able to compute the Taylor series of most other functions you encounter, without going through the tedious process of computing f (k) (0) for all k and then estimating the remainder. k x k 15