MATH 163 — R E V I E W E X a M 3 TOPICS COVERED (§12.2– §12.11) 1. Series ∑ an Know the Difference Between a Sequenc

MATH163—REVIEW EXAM 3

TOPICS COVERED ( 12.2– 12.11) § § ∞ 1. Series an nP=n0 Know the difference between a sequence and a series. ◦ Know definition of partial sums, definition of value of infinite series. ◦ ∞ What does it mean to say that a series a converges? Give examples of how a series ◦ k=k0 k can fail to converge. Does the convergence/divergenceP of a series depend on k0? ∞ What is a necessary condition on the sequence of summands a for the series a to ◦ n n=0 n converge? P Know examples of series that converge and series that diverge (for example, Harmonic, Alter- ◦ nating Harmonic, Geometric, and p-series). Be able to evaluate converging ◦ - geometric series - telescoping series - other series that happen to be known Taylor series for functions such as f(x) = 1/(1 x), ex, cos(x), or sin(x) at particular values of x. − Understand the meaning of absolute convergence and conditional convergence. ◦ Be able to determine whether a series converges absolutely or diverges using one of the following ◦ tests: Divergence Test, Integral Test, Direct Comparison Test, Limit Comparison Test, Ratio Test. (The integral test is not emphasized in this exam.) Suggestions: Always apply Divergence test first. I try to do a limit comparison next. Only try for a direct comparison if it is obvious how to establish the inequality. If above fails, Ratio test is good for powers and factorials. If above fails, try Integral test if you recognize function you can integrate. Alternatively: if you recognize the series to be a known Taylor series evaluated in its interval ∞ n of convergence, you are done. For example, you know n=0 1/2 converges because it is the Taylor series for 1/(1 x) (otherwise known as geometricP series) at x = 1/2, which is in the − ∞ interval of convergence. Similarly, you know that ( 10)n/n! converges. Why? In these n=0 − cases you not only know convergence, but the valueP they converge to! Know how to test for conditional convergence of an alternating series that does not converge ◦ absolutely, using the Alternating series Test. Note: Always check for absolute convergence first. Be able to estimate series by partial sums using the alternating series estimation test. Do you ◦ know of any other methods to estimate series?

∞ 2. Power Series c (x a)n, power series representation of functions n − nP=0 Find radius and interval of convergence of a power series. ◦ ∞ f (n)(a) 3. Taylor Series (x a)n. (Memorize this formula!) n! − nX=0 Note: The Taylor series of a function f about a base point x = a is simply its power series ◦ ∞ representation c (x a)n. It is unique and the formula for c is known. n=0 n − n P Be able to find Taylor series for any function about a given base point x = a. ◦ Memorize the MacLaurin series for ex, cos x, sin x, 1/(1 x) and their intervals of convergence. ◦ − Be able to find MacLaurin series from the memorized series using substitution, addition, mul- ◦ tiplication, differentiation or integration, and find their intervals of convergence. (In many cases you have to use a combination of these methods! For example, you need subsitution, integration and again substitution to obtain series for ln(1+ x2) about x = 0) Be able to approximate a function by a Taylor polynomial and estimate the error made using ◦ - Taylor’s Inequality. - the Alternating Series Estimation test, in those cases when you know that the Taylor series converges to f and it happens to be alternating. Be able to approximate functions that arise in in physics and engineering, in which there is a ◦ small parameter denoted by a variable not necessarily called x! Linearization: Memorize the formula for the linear approximation of f about x = a! (Note: it ◦ simply consists of the first two terms of the Taylor series.) Use it to approximate functions and changes of function values. What is the difference between the linearization and the tangent line?

PRACTICE PROBLEMS For each type of problem listed above, ﬁnd a problem in the book or homework like it. Do as many as you need to be comfortable. A good set of sample problems is given below. 1. REVIEW Chapter 12 (pp 794-796), Concept Check: 1,3,4,5(a-f),6,7,8,9,10,11 2. REVIEW Chapter 12 (pp 794-796), True False: 1-13,17-20.

3. For each of the following an: (i) Determine whether the sequences a converges or diverges. If it converges, ﬁnd the { n} limit. ∞ (ii) Determine whether the series n=1 an converges or diverges. P 1+ n3 9n+1 1 (a) a = (b) a = (c) a = n 1 + 2n3 n 10n n n (d) a = 2 + cos nπ (e) a = ln(n + 1) ln n n n − 4. Determine whether the following series converge absolutely, converge conditionally, or diverge. Make sure to clearly show all work. ∞ ∞ ∞ n sin n n2 + 1 (a) (b) (c) n3 + 1 1+ n2 n3 + 1 nX=1 nX=1 nX=1 ∞ ∞ ∞ ( 1)n ( 2)n ( 1)n ln n (d) − (e) − (f) − √n + 1 3n n nX=1 nX=2 nX=1 ∞ 1 (g) ( 1)n ln(2 + ) − n nX=1

5. Determine whether the following series converge or diverge. Evaluate those that converge. ∞ ∞ ∞ 22n+1 1 n 3 (a) (b) (c) ( 1)n+1 5n √ − 2n nX=1 nX=0 2 nX=2 ∞ ∞ ∞ ln n 2 1 n (d) (e) − (f) 1+ √n n(n + 1) n nX=1 nX=2 nX=1 ∞ ∞ ∞ 1 3n (g) ln (h) e−2n (i) ( 1)n n − n! nX=1 nX=0 nX=1 ∞ n! e2 e3 e4 (j) ( 1)n (k) 1 e + + . . . − 3n − 2! − 3! 4! − nX=1 6. REVIEW Chapter 12 (pp 794-796), # 33 7. REVIEW Chapter 12 (pp 794-796), # 35 8. REVIEW Chapter 12 (pp 794-796), # 38 9. REVIEW Chapter 12 (pp 794-796), # 45 ∞ xn 10. Let f(x) = . Find series representations for f ′, f ′′. Find the intervals of convergence n2 nX=1 of f, f ′, and f ′′. (Remember: the radius of convergence does not change under diﬀerentiation or integration! So, you only have to ﬁnd it for one of these series. However, the interval of convergence can change.) 11. Determine the radius and the interval of convergence of the following power series. ∞ ∞ ∞ ( x)n 3n(x + 4)n (a) ( 1)n4nxn (b) − (c) − n + 1 √n nX=0 nX=0 nX=1 12. REVIEW, # 47-52 13. REVIEW, # 55 14. REVIEW, # 56 15. REVIEW, # 60 16. Several important functions that arise in the mathematical sciences are given in terms of power series. An example is the Bessel function of order 0, given by

∞ ( 1)nx2n J (x)= − 0 22n(n!)2 nX=0

Find the interval of convergence of J0. 17. p 770, # 35 (show given series solve diﬀeq, show= ex) 18. Show that π π3/3!+π5/5! π7/7!+. . . converges to zero. How many terms must be computed − − to get within 0.01 of zero? 19. (a) Write down the linear approximation of a function f(x) about a base point x = a. (b) Write down the linear approximation of the change ∆f of a function f(x) from a base point a to a point a +∆x. (c) Use linear approximations to estimate the amount of paint needed to apply a coat of paint 0.05cm thick to a hemispherical dome with diameter 50 m. 20. The edge x of a cube was measured to be 30 cm with a possible measurement error of at most 0.1 cm. That is, x = 30cm +∆x , where ∆x 0.1 | |≤ Approximate the maximum error ∆y in using this measurement to compute | | (a) the volume of the cube (b) the surface area of the cube 21. The dosage D of diphenhydramine for a dog of body mass w kg, is D = kw2/3 mg, where k is a constant. A cocker spaniel has mass w = 10 kg according to a veterinarians scale. Use linear approximations to estimate the maximum allowable eror in w if the percentage error in the dosage D must be less than 5%. 22. (a) Find the ﬁrst three nonzero terms of the MacLaurin series for f(x)=(1+ x)k. (b) Find the linear approximation of f(x)=(1+ x)k. (c) Approximate √1+ x2 by a quadratic function if x is small.