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Page 1 Math 152, Spring 2008-copyright Joe Kahlig 1. (5 points) Find an equation of the sphere with center at (1, 4, −6) and passes through the point (−1, 3, 1) 2. (5 points) Let a =< 1, 0, −2 > and b =< −1, −2, −4 >. Compute the angle between a and b. 3. (5 points) Suppose that the nth partial sum of the series ∞ P an is given by sn = n=2 ∞ P 5n + ln(n) . Will the series n an will converge or diverge? If the series converges, find the sum of the series if possible. n=2 4. (14 points) Find the radius and the interval of convergence for ∞ X n=2 nxn + 1) 4n (n2 5. (5 points) For the Taylor series for g(x) = ln(x + 4) centered at a = −2, find the coefficient of the (x + 2)3 term? 6. (8 points) Use the series to approximate this integral so that the error < 10−2 . Use the fewest number of terms possible to attain the prescribed accuracy. Z0.5 3 e−x dx = 0 7. (16 points) Find the power series representation for these functions centered at a = 0. Also give the radius of convergence. sin(2x) x2 4 (b) g(x) = 2 9x + 4 (a) f (x) = 8. (14 points) Determine if these series converge or diverge. If it converges, find its sum. Fully justify your answer. (a) n ∞ n X −1 1 −4 7 5 9 n=0 (b) ∞ X 1 (n + 1)(n + 2) n=5 9. (14 points) Determine if the series is divergent, absolutely convergent, or just convergent. Fully justify your answer. (a) ∞ X (−1)n √ n n=1 (b) ∞ X (2n)! n!(n + 1)! n=1 10. (14 points) Determine if these series converge or diverge. Fully justify your answer. ∞ X ln(n) (a) n3 n=1 (b) ∞ X (−1)n n √ n2 + 5 n=1 Check the back of the page for more problems.