MA137 Mathematical Analysis

1. Evaluate 1X n=1 5 ?? 3n 23n . 2. Prove that, for n > 2, 1 + 1 n n2 > n2 ?? 1 2 . Deduce that 1X n=1 1 + 1 n ??n2 converges. 3. Show that log( 3 2 ) 6 1X n=1 1 n(2n + 1) 6 1 3 + log( 3 2 ): 4. Let (bn)n2N+ be a bounded sequence. Prove that if P1 n=1 an is absolutely convergent then P1 n=1 anbn is also absolutely convergent. 5. Sketch the graph of f(x) = p x; x > 0, and use it, together with relevant calculations, to explain why 2 3N3=2 < XN n=1 p n < 2 3 (N3=2 ?? 1) + p N: 6. Find the value of 1X n=2 1 n3 ?? n by the telescoping series method. 7. Recall from Question 1 on Examples Sheet 0 that, 8 a; b 2 R, jabj 6 1 2 (a2 + b2): (1) (a) Suppose that an > 0 8 n 2 N+ and that P1 n=1 an < +1. Make appropriate choices of a and b in (1) to prove that 1X n=1 p an n < +1: (b) Provide an example of a sequence (an)n2N+ for which an > 0 8 n 2 N+ and P1 P n=1 an < +1 but 1 n=1 p an = +1. Worked Example Find the value of 1X n=1 p n + 2 ?? p n p n(n + 2) by the telescoping series method. Solution p n + 2 ?? p n p n(n + 2) = 1 p n ?? 1 p n + 2 . Therefore, p 3 ?? p 1 p 3 = 1 1 ?? 1 p 3 p 4 ?? p 2 p 8 = 1 p 2 ?? 1 p 4 p 5 ?? p 3 p 15 = 1 p 3 ?? 1 p 5 ... ... p k + 1 ?? p k ?? 1 p (k ?? 1)(k + 1) = 1 p k ?? 1 ?? 1 p k + 1 p k + 2 ?? p k p k(k + 2) = 1 p k ?? 1 p k + 2 : Therefore, Xk n=1 p n + 2 ?? p n p n(n + 2) = 1 + 1 p 2 ?? 1 p k + 1 ?? 1 p k + 2 : It follows that 1X n=1 p n + 2 ?? p n p n(n + 2) = lim k!1 1 + 1 p 2 ?? 1 p k + 1 ?? 1 p k + 2 = 1 + 1 p 2 : 17 Invitational Speech This speech will be a three-part speech: the first part is the issue; the second part is your dialogue with the audience and the third part is your summary and your conclusion. Criteria for the speech: 6-7 minutes for the issues part of the speech; 2-3 minutes for the dialogue and 1-2 minutes for the conclusion. The speech should contain three different sources in it. Only two of the sources may be taken from the Internet. ( "An outline and an annotated bibliography are required for this speech. This is a capstone speech which utilizes the rhetorical concepts.") THIS ALL I NEED. I need about 5 great slides ppt Poverty: America’s crippling problem