This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: n →∞ a n = c exists we have that c = c + 90 10 or c = 10. Thus we expect that the sequence increases towards 10. a) The sequence is bounded above by 10. To see this we use induction. It is true ±or a 1 since a 1 = 5 < 10. Assume it is true ±or some n , i.e., a n ≤ 10. Then a n +1 ≤ 10 + 90 10 = 10 1 which proves what we have claimed. b) Next we check that a n is increasing. Calculate a n +1a n = a n + 90 10a n = 9 10a n 10 ≥ since a n ≤ 10. c) c = 10. d) Next, note that ≤ 10a n +1 = 10a n + 90 10 = 10a n 10 = 1 10 (10a n ) . Thus ≤ 10a n +1 = ± 1 10 ² n (10a 1 ) = 5 ± 1 10 ² n . Thus if n = 7 we get that 5107 < 106 and hence a n < 106 for all n ≥ 8. 2...
View
Full Document
 Fall '08
 N/A
 Calculus, Logic, lim

Click to edit the document details