Test 7

Pure Mathematics – Sequences

p.1

King’s College 2007 – 2008 F.6 Pure Mathematics Revision Test 7 Time Allowed: 40 minutes Total Mark: 30 1.

(2003) Let {xn} be a sequence of positive real numbers, where x1 = 2 and xn+1 = xn2 – xn + 1 for all n = 1, 2, n 1 3, …. Define S n = ∑ for all n = 1, 2, 3, …. i =1 x i (a) Using mathematical induction, prove that for any positive integer n, (i) x n > n (ii) S n = 1 −

1 x n +1 − 1

.

(b) Using (a), or otherwise, prove that lim S n exists. n→∞

(8 marks) 2.

(2005) (a) By considering the function f(x) = x – ln(x + 1), or otherwise, prove that x ≥ ln ( x + 1) for all x > -1 ∞

(b) Using (a), prove that the series

1

∑n

is divergent.

n =1

(7 marks) 3.

(2005) Let a1 and b1 be real numbers satisfying a1b1 > 0 . For each n = 1, 2, 3, …, define a n + bn a n + bn 2

a n +1 =

2

and bn +1 =

2a n bn a n + bn

(a) Suppose a n ≥ bn > 0 . (i) Prove that a n ≥ bn for all n = 1, 2, 3, …. (ii) Prove that the sequence {an} is monotonic decreasing and that the sequence {bn} is monotonic increasing. (iii) Prove that lim a n and lim bn both exist. n →∞

n→∞

(iv) Prove that lim a n = lim bn . n →∞

n →∞

(v) Find lim(a n + bn ) and lim a n in terms of a1 and b1. n →∞

n →∞

(b) Suppose a n ≤ bn < 0 . Do the limits of the sequences {an} and {bn} exist? Explain your answer. (3 marks) --End of Test--