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# Ii converges sum of two convergent geometric series

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Unformatted text preview: on required. 3.5 a) Which of the following correctly classiﬁes the series i) Diverges: limn→∞ an = 0. ii) Converges: sum of two convergent geometric series. iii) Converges: Comparison test with geometric series with ratio 3/e. iv) Converges: Comparison test with geometric series with ratio e/4. v) Diverges: constant multiple of the harmonic series. ∞ n3 b) Which of the following correctly classiﬁes the series and gives a 4 n=1 5 + n valid reason? Circle your answer. No other reason required. i) Diverges: limn→∞ an = 0. ii) Converges: Ratio test. iii) Converges: Comparison test with a p-series, p > 1. iv) Diverges: Ratio test. v) Diverges: Comparison or limit comparison with the harmonic series. 3.6 Determine if the following series converge or diverge. Give your reasoning using complete sentences. MTH 142 Exam 3 Practice Problems Spring 2011 a) ∞ n=1 b) (−1)n ∞ n=2 3 n n+2 (−1)n+1 1 ln n 3.7 a) Find the interval of convergence for the power series b) Find the radius of convergence for ∞ 1 (x − 3)n n n=1 n2 ∞ n! n x n=1 (2n)! 3.8 a) Find the Taylor series about x = π of cos(x). Your answer should clearly indicate the pattern of the terms. (Note the center, π .) b) Find the Taylor series of ln(x) at x = 1. 3.9 Give the Taylor polynomial at 0 of degree 4 for each of the following functions. Use the easiest method you know, including your knowledge of the Taylor series for sinx, ex , (1 + x)p , etc. a) f (x) = x2 cos(x). b) g (x) = √ c) f (x) = 1 1 − 2x x 0 sin(t2 ) dt d) f (x) = (1 + x) sin(2x) 3.10 The graph of y = f (x) passes through the point (0, −2) and near that point the function is decreasing and concave up. If the third degree Taylor polynomial of f near 0 is P2 (x) = a + bx + cx2 what are the signs of a, b and c? 2 b) If f (x) = e−x ﬁnd f (8) (0) and f (9) (0), that is, the 8th and 9th derivative of f at 0. Suggestion: Consider the Taylor series of f and look at the coeﬃcients of x8 and x9 ....
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## This note was uploaded on 02/04/2014 for the course MAT 132 taught by Professor Poole during the Fall '08 term at SUNY Stony Brook.

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