4. (11 points) rI'he following parts are not related. (a) Place the following in increasing order (from smallest to largest). You do not need to give expla nations. 1 1 A _ Z 2k+3k D '22? k=100 k:1 2k+3k _ 2 E k:] k:IUUU C =3 (b) Determine whether each of the following series converges or diverges. Write up your answer carefully. (Whatever test you use, you must explicitly state the conditions that are present that allow you to use that test. Your logic should be clear, easy to follow, and carefully written up.) DO . 3k2+mk+e .. (1)k sin(.-'c) 1' 6k4+3k2+1000k 11' Z \"2' \"1' E 1.01k' 5. (6 points) Decide whether the following statements are true or false. Explain why. (a) Let Sn he the nth partial sum of the series Z: ck, that is Sn = {11 +0.2 +1113 +- - -+an. If Sn = 35:] a m then the series 2 ck must diverge. k=l DC DC 00 a (b) If E on converges and 2 bn diverges, and b % D for all n, then the series bn also converges. 1 11 11:1 11:] n: 00 (c) Suppose that the power series Zak (I + 3):\" converges and its radius of convergence is 2, then k:l 00 the series E a]; must converge. .lc:l 1 1+:c4' l. (7 points) Consider the function f (I) = 1 1 + 2:4 and - - - notation (with at least three terms written out). What's the radius of convergence of the power series? (a) Find the power series representation of centered at 0. Write your answer in both sigma 1 1 + 1:4 notation (with at least three terms written out). What's the largest open interval of convergence of the power series? (b) Find a power series representation of / dz. Write your answer in both sigma and - -- 1 1 + r4 notation (with at least three terms written out). 0.1 (c) Write down a series that converges to / dz. Write your answer in both sigma and - -- 0 1 1+:r4 o 1 (d) Use the result in part (0), how many terms does one need in order to estimate / 0 with an error of magnitude less than 1010? Explain your reasoning. 2. (6 points) Determine the radius of convergence and the interval of convergence for the following power series. If the interval has endpoints, don't forget to check those too. i (4%) (a: + 2)'= k=1 3. (4 points) In this question, we will nd the degree n Taylor polynomial for the function f (x) = 111(1 +x) centered at 1. (3.) Find f(1), f'(1), f\"(1), f(3)(1), and fik)(1) where k is any positive integer. (b) Find the degree n Taylor polynomial for f (as) = ln(1 + 2:) centered at 1. Write your answer in - -- notation. (Write down at least the rst three terms + - - - + the last term.)