107-Sec Math 250 Spring 2003 No. score 1(a, b) 1(c)
Exam 3
Name: Recitation Instructor:
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1. (30 points, 10 points each) Determine whether the following series converge absolutely, converge conditionally or diverge. You must show all details to receive credit.
a.
k=1
(-1)k
3k 3 - 1 k 4.07 + 2
b.
k=1
(-1)
k
4k - 1 3k + 2
k
1
c.
k=1
(-1)k
99k k! (2k + 1)!
2. (18 points) Find the interval on which the following power series converges absolutely. Also, find the radius of convergence and make sure to discuss in details the convergence/divergence of the series at the end points of the interval you have found.
k=1
(-1)k (x - 2)k+1 k
(1)
2
3. (9 points) Find the first three non-zero terms of the Taylor series of f (x) = (1 + x) 1/3 about x = 0. You need not find every term in the series.
1 4. (8 points) By using Taylor the series: xk , |x| < 1; find the Taylor series of = 1-x k=0 x5 f (x) = about x = 0. Make sure to include the interval of convergence. 9 + x2
5. (8 points) By eliminating the parameter, find the xy-equation for the curve C : x = 2 - cos t, y = -1 + sin t, 0 t 2; and sketch C with orientation.
3
6. (12 points) Suppose that a function f (x) has the following Taylor series about x = 0:
f (x) =
k=3
(-1)k 2k x , -2 < x < 2. 4k (k + 1)
(2)
a. (8 pts.) Find the exact value of f (66) (0)...