Sequence - Assignment

Pure Mathematics – Limit of Sequence

p.1

Limit of Sequence Assignment n +1 1 1 . and y n = ∑ k =1 n + k k =1 n + k is strictly increasing and that the sequence n

1.

For every positive integer n, define x n = ∑

(a) Prove that the sequence {x n } {y n } is strictly decreasing. (b) Prove that the sequence {x n } and 2.

{y n }

converge to the same limit.

Let {a n } be a sequence of positive real numbers, where a1 = 1 and a n =

12a n −1 + 12 , n = 2, 3, 4, …. a n −1 + 13

(a) Prove that a n ≤ 3 for all positive integers n. (b) Prove that {a n } is convergent and find its limit. 3.

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 =

2 a n bn . a n + bn

(a) Suppose a1 ≥ b1 > 0 . (i) Prove that a n ≥ bn for all n = 1, 2, 3, …. (ii) Prove that the sequence {a n } 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 an in terms of a1 and b1. n→∞

n →∞

(b) Suppose a1 ≤ b1 < 0 . Do the limits of the sequences {a n } and {bn } exist? Explain your answer.

Pure Mathematics – Limit of Sequence

4.

p.2

Let {x n } be a sequence of real numbers such that x1 > x2 and 3xn+2 – xn+1 – 2xn = 0 for n = 1, 2, 3, …. 2 n −1 (x1 − x2 ) . 3n (ii) Show that sequence {x1 , x3 , x5 , K} is strictly decreasing and that the sequence {x 2 , x 4 , x6 ,K} strictly increasing.

(a) (i)

Show that for n ≥ 1 , x n + 2 − x n = (− 1) ⋅ n

(b) (i) For any positive integers n, show that x2n < x2n-1. (ii) Show that the sequences {x1 , x3 , x5 ,K} and {x 2 , x 4 , x6 ,K} converge to the same limit. p

(c) By considering

∑ (x n =1

n+2

− x n ) or otherwise, find lim x n in terms of x1 n →∞

and x2. [You may use the fact, without proof, that from (b)(ii), lim x n exists.] n →∞