Reference no: EM132607893

Question 1. Prove that if lim_n→∞ x_n/n = l ≠ 0 then the sequence (x_n) is not bounded.

Question 2. Is the sequence {(1 + 1/n)ⁿ}^∞_n=1 convergent? If yes, denote lim_n→∞(1 + 1/n)ⁿ = e and prove that lim_n→∞(1 + 2/n)ⁿ = e².

Question 3. If x_n > 0 for all n ∈ N and lim_n→∞ x_n+1/n = l then show that

(i) if l < 1 then lim_n→∞ x_n = 0

(ii) if l > 1 then lim_n→∞ x_n = ∞

Question 4. Let x₁ > x₂ > 0 and x_n+1 = 1 (x_n + x_n-1) for n ≥ 2. Show that

(i) ( x_2n-1) is a non-increasing subsequence of (x_n)
(ii) ( x_2n) is a non-decreasing subsequence of (x_n)
(iii) ( x_n) converges to 2

Question 5. Let ( x_n) be a sequence of real numbers. Let |x_n+1 - x_n| ≤ crⁿ, ∀ n ∈ N, for some c ∈ R and 0 < r < 1. Then show that (x_n) is a convergent sequence.

Question 6. Let ( x_n) be defined as x₁ = 1, x_n+1 = 1 + 1/x_n for all n ∈ N. Then show that (x_n) is a Cauchy sequence.

Question 7. Let ( s_n) be defined as sn = 1 + 1/2 + 1/3 + .... + 1/n for all n ∈ N. Prove that (sn) is not a Cauchy sequence.

Question 8. Find the limit superior and the limit inferior of the following sequences

(i)1, 2, 3, 1, 2, 3, 1, 2, 3, .....
(ii) {sin(nΠ/2)}^∞_n=1
(iii) {(1 + 1/n) cos nΠ}^∞_n=1
(iv) {(1 + 1/n)ⁿ}^∞_n=1
(v) {(1 + (-1)ⁿ + 1/2ⁿ)^1/n }∞