Math 107 Fall ’12  Exam 2  Score:
Name:  TA’s Name:

Instructions: You must show supporting work to receive full and partial credits. No text book, notes, formula sheets allowed.

1(20pts) Find the following for the power series: (a) center and radius of convergence; (b) interval of absolute convergence; (c) points of conditional convergence if any, (d) interval of convergence; (e) intervals of divergence,

∑_{n=0}^{∞} \frac{1}{2^{n}\sqrt{n + 1}}(x + 2)^{n}

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2(10pts) A rectangular swimming pool 50 ft long, 20 ft wide, and 10 ft deep is filled with water to a depth of 9 ft. Set up an integral to find the work required to pump all of it over the top rim. (The weight density of water is 62.4 lb per ft^{3}.)

3(10pts) (a) Find the Taylor polynomial P_{3}(x) of of f(x) = ln x at point x = a = 1.

(b) Compare the values of ln(1.1) and P_{3}(1.1) by stating the decimal place to which these two values agree.

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4(10pts) (a) Use the Integral Test to determine the value of p such that ∑_{n=2}^{∞} \frac{1}{n(ln n)^{p}} converges. 5(20pts) Determine if the following series converge.

(a) ∑_{n=1}^{∞} cos(\frac{1}{n})

(b) ∑_{n=0}^{∞} \frac{2^{n}n!}{(2n)!}

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6(20pts) Use comparison tests to determine if the following series converge absolutely or diverges:

(a) ∑_{n=0}^{∞} \frac{2^{n} sin n}{3^{n} + 2}

(b) ∑_{n=1}^{∞} \frac{n + 1}{\sqrt{n(n^{2} + 1)}}

7(10pts) If you use the first 9 terms of the series ∑_{n=1}^{∞} \frac{(−1)^{n−1}}{n^{2}} to approximate it, how many decimal place

accuracy do you expect to get for the approximation? Explain your work. 2 Bonus Points: True or False: There are 12 schools in the Big XII Conference. (... The End)