1873_solutions

When 1 and is divergent when 2 a n 1 an 1 n 1 a

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Unformatted text preview: aw that the series  n is conditionally convergent when 1    0 and that   1n  n n for each  and n. We can summarize these facts as follows: If  is a nonnegative integer then the given series converges for all x. If  is not an integer and   0 then the series converges absolutely when |x | |x |  1. 1 and diverges if If  1. 1 then the series converges absolutely when |x |  1 and diverges if |x | If 1    0 then the series converges absolutely when |x |  1, converges conditionally when x  1 and diverges when either x 1 or x  1. 8. Prove that if x is not an integer multiple of 2 then Ý j1 sin jx j 345 1 x sin 2 Hint: Use the inequality obtained after Dirichlet’s test. 9. Prove Abel’s test for convergence of a series which states that if a n is a decreasing sequence of positive numbers and if b n is a convergent series then the series a n b n is convergent. This theorem may be proved by the method of proof of Dirichlet’s test but it also follows very simply from the statement of Dirichlet’s test. Which proof do you prefer? We show the proof that uses the statement of Dirichlet’s test. Assume that lim a  a. nÝ n From Dirichlet’s test we know that an a bn is convergent. Therefore, since the series ab n is convergent and anbn  an a b n  ab n for each n, it follows that anbn converges. 10. Give an example of a sequence a n of positive numbers and a sequence b n of real numbers such that each of the following conditions holds: a. We have a n 0 as n b. The sequence of number c. The series We define b n  Ý. n b j1 j is bounded. a n b n is divergent. 1 n for each n and we define 1 n if n is even 1 2n an  if n is odd 11. Give an example of sequences a n and b n such that the following conditions hold: a. The sequence a n is a decreasing sequence of positive numbers. b. The sequence of number c. The series We define b n  n b j1 j is bounded. a n b n is divergent. 1 n and we define an  1  1 n for each n. Some Exercises on Products of Series 1 n x n and x n . By looking at the sums of these three series, 1. Calculate the Cauchy product of the series verify that Cauchy’s theorem is true for these series when |x |  1. We know that 346 Ý 1 1x 1 nxn  n0 and Ý 1 xn  n0 1 nxn The nth term of the Cauchy product of 1 nj nj j xx x n if n is even  x 2n . We observe that and so the Cauch product is x 2n n0  1 1 if n is odd 0 j0 Ý x n is and n 1 x x2  1 1x Ý 1 1  x Ý 1 nn x xn . n0 n0 2. This exercises requires a knowledge of the binomial theorem. Show that the Cauchy product of the two x  y n /n!. As you may know, the sums of these series are e x and e y and series x n /n! and y n /n! is xy respectively, and you will see this fact officially in a later subsection. What does Cauchy’s theorem say e for these three series? The nth term of the Cauchy product of x n /n! and y n /n! is n j0 xn j n j! yj j! n  xn jyj n j0 n! j !j! 1 n! n 1 n!  xn jyj j0 n j  xy n! n x  y n /n! converge, respectively, to e x...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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