Geometry

Topology

Monotonic Sequences

         

Suppose {an}\{a_n\} is a sequence of real numbers such that an+1=12an+3a_{n+1} = \frac{1}{2}a_n + 3 for each natural number n1.n \geq 1. Which of these statements are true?

A. If limnan=L,\displaystyle \lim_{n \to \infty}a_n = L, then L=12L+3.L = \frac{1}{2}L + 3.

B. limnan=6.\displaystyle \lim_{n \to \infty}a_n = 6.

Suppose {an}\{a_n\} is a sequence of real numbers such that a13a_1 \neq -3 and an+1=2an+3a_{n+1} = 2a_n + 3 for each natural number n1.n \geq 1. Which of these statements are true?

A. If limnan=L,\displaystyle \lim_{n \to \infty}a_n = L, then L=2L+3.L = 2L + 3.

B. limnan=3.\displaystyle \lim_{n \to \infty}a_n = -3.

Suppose {an}\{a_n\} is a sequence of real numbers such that an+1=12an+3a_{n+1} = \frac{1}{2}a_n + 3 for each natural number n1.n \geq 1. Which of these statements are true?

A. {an}\{a_n\} is bounded.

B. {an}\{a_n\} is monotone.

C. {an}\{a_n\} converges.

Definitions. A sequence {xn}\{x_n\} of real number is:

bounded if there is a real number MM such that M<xn<M-M < x_n < M for all n.n.

monotone if xnxn+1x_n \leq x_{n+1} for all n,n, or xnxn+1x_n \geq x_{n+1} for all n.n.

convergent if there is a number LL such that for every ϵ>0,\epsilon > 0, there is a natural number NN satisfying

n>NxnL<ϵ.n > N \Rightarrow |x_n - L| < \epsilon.

Suppose {an}\{a_n\} is a sequence of real numbers such that a13a_1 \neq -3 and an+1=2an+3a_{n+1} = 2a_n + 3 for each natural number n1.n \geq 1. Which of these statements are true?

A. {an}\{a_n\} is bounded.

B. {an}\{a_n\} is monotone.

C. {an}\{a_n\} converges.

Definitions. A sequence {xn}\{x_n\} of real number is:

bounded if there is a real number MM such that M<xn<M-M < x_n < M for all n.n.

monotone if xnxn+1x_n \leq x_{n+1} for all n,n, or xnxn+1x_n \geq x_{n+1} for all n.n.

convergent if there is a number LL such that for every ϵ>0,\epsilon > 0, there is a natural number NN satisfying

n>NxnL<ϵ.n > N \Rightarrow |x_n - L| < \epsilon.

Which of these statements are true?

A. Every bounded, monotone sequence of real numbers converges

B. Every bounded sequence of real numbers converges

C. Every monotone sequence of real numbers converges

Definitions. A sequence {xn}\{x_n\} of real number is:

bounded if there is a real number MM such that M<xn<M-M < x_n < M for all n.n.

monotone if xnxn+1x_n \leq x_{n+1} for all n,n, or xnxn+1x_n \geq x_{n+1} for all n.n.

convergent if there is a number LL such that for every ϵ>0,\epsilon > 0, there is a natural number NN satisfying

n>NxnL<ϵ.n > N \Rightarrow |x_n - L| < \epsilon.

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