Ii converges sum of two convergent geometric series

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Ask a homework question - tutors are online