M for all n ≥ 1

HOMEWORK 1

A sequence (x_n) is bounded if there exists M > 0 such that |x_n| ≤ M for all n ≥ 1.

(1) Let (xn) be the sequence defined by xn= 1

1² + 1

2² + · · · + 1 n². Show that (x_n) is convergent:

(a) Show that (xn) is increasing.

(b) Show that (x_n) is bounded. (Hint: n²> n(n − 1), for all n ≥ 2.) (2) Let (xn) be the sequence of real numbers defined by

xn+1 =√

2 + xn, x1=√ 2.

(a) Show that (x_n) is increasing. (Use induction).

(b) Show that x_n< 2 for all n ≥ 1. (Use induction).

(3) Find the limit of (an) (a) a_n= 1 − 5n⁴

n⁴+ 8n³. (b) a_n= √ⁿ

2ⁿ+ 3ⁿ.

(4) Find the sum of the following series:

(a)

∞

X

n=1

2n + 1 n²(n + 1)². (b)

∞

X

n=1

4

(4n − 3)(4n + 1).

(5) Suppose that x₀ = 1 and x₁= 2. Define x_n= x_n−1+ x_n−2

2 , n ≥ 2.

Compute lim

n→∞x_n (6) Let x₀= 1. Define

xn= 1 + 1 2 + xn−1

, n ≥ 1.

Suppose that we know (xn) is convergent. Find lim

n→∞xn. (7) Suppose xn= cosx

2cos x

2²· · · cos x

2ⁿ. Find lim

n→∞xn. (8) Let x_n=

(n+1)²

X

k=n²

√1 k. (a) Show that

2√

k + 1 − 2

√ k < 1

√ k < 2

√

k − 2√ k − 1 for all k ≥ 1.

(b) Use (a) to compute lim

n→∞xn.

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