Use definition to show that lim n→∞ an bn = a b

1. hw 2 All the sequences are sequences of real numbers.

(1) Show that the sequence (an) defined below is a Cauchy sequence an= cos 1!

1 · 2 +cos 2!

2 · 3 + · · · + cos n!

n(n + 1), n ≥ 1.

(2) Let (an) and (bn) be convergent sequences. Denote lim

n→∞an = a and lim

n→∞bn = b.

Suppose b 6= 0. Use definition to show that lim

n→∞

an

b_n = a

b. (Hint: use definition of limit to show that there exists a natural number N such that |b_n| > |b|/2 for n ≥ N.

Hence |bnb| > b²/2 for n ≥ N.)

(3) (Sandwich Principle) Suppose (a_n), and (b_n) and (c_n) are sequences. Suppose that there exists N such that an ≤ b_n ≤ c_n for n ≥ N. If both of (an) and (cn) are convergent to a, show that (b_n) is also convergent to a.

(4) Evaluate the following limits. You can either use definition of limit or theorem to show that the sequences given below are convergent. When you evaluate limits using theorems or propositions proved in class, please indicate the one you are using.

(a) lim

n→∞

n!

nⁿ. (b) lim

n→∞

√n

2ⁿ+ 3ⁿ. (c) lim

n→∞(√

n + 1 −√ n).

(d) lim

n→∞

1 + a + · · · + aⁿ

1 + b + · · · + bⁿ where |a| < 1, |b| < 1.

(e) lim

n→∞

√3

n²sin(n!) n + 1 .

(5) Let (an) and (bn) be sequences. Give a counterexample to each of the following equalities. Here we do not assume the convergence of (a_n) and (b_n).

(a) lim

n→∞(kan) = k · lim

n→∞an. Here k is a real number.

(b) lim

n→∞(an+ bn) = lim

n→∞an+ lim

n→∞bn. (c) lim

n→∞(a_nb_n) = lim

n→∞a_n· lim

n→∞b_n. (d) lim

n→∞

a_n bn

= lim_n→∞a_n limn→∞bn

.

(6) Let (an) and (bn) be two convergent sequences. Let a and b be their limit respec- tively. Suppose that there exist N ∈ N so that an ≤ b_n for all n ≥ N. Show that a ≤ b.

(7) Let a₁, · · · , a_k be positive numbers and M = max{a₁, · · · , a_k} the maximum of a₁, · · · , a_k. Prove or disprove lim

n→∞

pan ⁿ₁ + · · · + aⁿ_k = M. (See 4, (b).) (8) Let (an) be the sequence defined by

an+1 =√

2 + an, n ≥ 1 with a1 =√

2.

1

2

(a) Use mathematical induction to show that (a_n) is increasing and bounded above.

(b) Use (a), we know the sequence (an) is convergent by the monotone sequence properties. Find the limit of (a_n).

(9) Let (x_n) be a sequence defined by x_n= 1 + 1

1!+ · · · + 1

n!, n ≥ 1.

(a) Show that (x_n) is a Cauchy sequence.

(b) Using the completeness of R, (xn) is a convergent sequence. Denote its limit by e. Define a sequence (p_n) by p_n = n!x_n, for n ≥ 1. Then p_n are all natural numbers. Show that |n!e − pn| > 0 for n ≥ 1.

(c) Show that e is an irrational number. (Hint: use homework 1 exercise 4.)