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Sol-HW3 - Math Methods Fall 2010 Solutions for Assignment 3...

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Math Methods Solutions for Assignment 3 Sep. 08, 2010 Fall 2010 Infinite Series Due Sep. 15, 2010 1 (10). Determine if the following series converge: X n =10 1 n (ln n )(ln(ln n )) Since this is a series of positive, monotone decreasing terms, the sum converges or diverges with the integral, Z 10 1 x ln x ln(ln x ) dx = Z ln(10) 1 y ln y dy = Z ln(ln 10) 1 z dz . Since the integral diverges, the series also diverges. X n =2 ln n ln n X n =2 1 n ln(ln n ) X n =1 1 n The sum is divergent by the comparsion test. X n =0 4 n + 1 3 n - 2 Since the terms in the sum do not vanish as n → ∞ , the series is divergent. X n =1 ( n !) 2 ( n 2 )! Applying the ratio test we get lim n →∞ a n +1 a n = lim n →∞ (( n + 1)!) 2 ( n 2 )! (( n + 1) 2 )!( n !) 2 = lim n →∞ ( n + 1) 2 (( n + 1) 2 - n 2 )! = lim n →∞ ( n + 1) 2 (2 n + 1)! = 0 The series is convergent. X n =0 (Log π 2) n This is a geometric series. Since | Log π 2 | < 1, the series converges. 2 (6). Find the circle of convergence of the following series: X n =0 nz n Radius of convergence is R = lim n →∞ n n + 1 = lim
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