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 n≥1.n \geq 1. Which of these statements are true?

A. If lim⁡n→∞an=L,\displaystyle \lim_{n \to \infty}a_n = L, then L=12L+3.L = \frac{1}{2}L + 3.

B. lim⁡n→∞an=6.\displaystyle \lim_{n \to \infty}a_n = 6.

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by Brilliant Staff

Suppose {an}\{a_n\} is a sequence of real numbers such that a1≠−3a_1 \neq -3 and an+1=2an+3a_{n+1} = 2a_n + 3 for each natural number n≥1.n \geq 1. Which of these statements are true?

A. If lim⁡n→∞an=L,\displaystyle \lim_{n \to \infty}a_n = L, then L=2L+3.L = 2L + 3.

B. lim⁡n→∞an=−3.\displaystyle \lim_{n \to \infty}a_n = -3.

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by Brilliant Staff

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 n≥1.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 xn≤xn+1x_n \leq x_{n+1} for all n,n, or xn≥xn+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>N⇒∣xn−L∣<ϵ.n > N \Rightarrow |x_n - L| < \epsilon.

Show explanation
by Brilliant Staff

Suppose {an}\{a_n\} is a sequence of real numbers such that a1≠−3a_1 \neq -3 and an+1=2an+3a_{n+1} = 2a_n + 3 for each natural number n≥1.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 xn≤xn+1x_n \leq x_{n+1} for all n,n, or xn≥xn+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>N⇒∣xn−L∣<ϵ.n > N \Rightarrow |x_n - L| < \epsilon.

Show explanation
by Brilliant Staff

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 xn≤xn+1x_n \leq x_{n+1} for all n,n, or xn≥xn+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>N⇒∣xn−L∣<ϵ.n > N \Rightarrow |x_n - L| < \epsilon.

Show explanation
by Brilliant Staff
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