Math 323: Sequences and Series Homework Solutions, Assignments of Mathematics

Download Math 323: Sequences and Series Homework Solutions and more Assignments Mathematics in PDF only on Docsity! Math 323 Homework # 4 – Due Wednesday July 28, 2004 Partial credit is possible, but you must show all work. Name: I hereby testify that this is individual work. Signed: 1. Consider the sequence defined recursively by x1 = 1; xn+1 = xn + 1 + xn 1 + 2xn . (a) Use induction to show that for all n ∈ N, xn > 0. Use the result to deduce that xn+1 > xn (b) Show that for all n ∈ N; xn+1 > xn + 1 2 . Deduce that xn > 1 + n− 1 2 . (c) Show that limn→∞ xn =∞. 2 2. Consider the sequence x1 = 9; xn+1 = x2n + 9 2xn . (a) Show that for all n ∈ N, xn > 3. (b) Show that xn+1 − 3 = (xn − 3)2 2xn and xn+1 + 3 = (xn + 3) 2 2xn (c) Let yn = xn − 3 xn + 3 . Express yn+1 in terms of yn. (d) Show that yn = ( 1 2 )2n−1 . 5 (d) Find an explicity formula for ∑n k=1 1 4k2 − 1 . (e) What is ∑∞ k=1 1 4k2 − 1 . 6. Let (xn)n∈N be an arithmetic sequence. Let q > 0 be given. Show that yn = q xn is a geometric sequence. 6 7. Consider the sequence x1 = 1; xn+1 = 1 2 ( xn + p xn ) , where p > 1 is given. Show that for all n ∈ N, 1 ≤ xn < p + 1. 8. Let x1 = −2, x2 = 5 and xn = 1 6 (xn−1 + xn−2) for n ≥ 3. (a) Consider the sequence yn = 2xn − xn−1 for n ≥ 2. Show that (yn)n≥1 is a geometric sequence. Express it as a function of n. 7 (b) Let tn = xn+1 + 1 3 xn for n ≥ 1. Show that (tn)n≥1 is also a geometric sequence. What is its expression.