(1) Which of the following sequences are bounded? (a) an= (−1)n n + 1 for n ∈ N in R

1. Quizz 5 A sequence of real numbers (a_n) is called

(1) nondecreasing (increasing) if an≤ a_n+1 (an< an+1);

(2) nonincreasing (decreasing) if a_n≥ a_n+1 (a_n> a_n+1.)

The sequence (an) is monotone if it is either nondecreasing or nonincreasing.

A sequence (a_n) in R^p (p ≥) is said to be bounded if there exists M > 0 so that ka_nk_R^p ≤ M for any n ≥ 1.

Theorem 1.1. (Monotone Sequence Property) A bounded monotone sequence in R is convergent.

(1) Which of the following sequences are bounded?

(a) an= (−1)ⁿ

n + 1 for n ∈ N in R.

(b) a_n= n²+ n for n ∈ N in R.

(c) an= (cos√

n, sin√

n) for n ∈ N in R². (d) an=



(−1)ⁿ,1

n, cos(n!πα)



for n ∈ N in R³ Here α is an irrational number.

(e) an= 3n − 1

3n + 2,4n²+ 1 n²+ 5



for n ∈ N in R².

(2) Use -definition to show that the sequence (xn) is convergent. Here x_n= 2n²+ 3

3n²+ 1, n ∈ N.

Guess that its limit is 2/3. For any  > 0, find N_ so that 

2n²+ 3 3n²+ 1−2

3    

< 

holds for any n ≥ N_. Usually, we choose N_ to be the smallest integer such that the above equation holds.

(3) Let x₁∈ R with x1> 1. Define (x_n) by x_n+1 = 2 − 1

xn for n ∈ N.

Show that the sequence is monotone and bounded. What is its limit?

(4) Let a > 0 and z₁> 0. Define a sequence (z_n) by z_n+1 =√

a + z_n, for n ∈ N.

Show that (zn) is convergent and find its limit.

(5) Determine the convergence/divergence of the sequence (x_n) defined by xn= 1

n + 1+ 1

n + 2 + · · · + 1 2n =

n

X

i=1

1 n + i. (6) Let (x_n) be a sequence of real numbers defined by

x_n+1= x_n+ (−1)ⁿ (2n + 1)!

with x₁= 1. Test the convergence/divergence of (x_n).

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