You must be logged in to see worked solutions.

Already have an account? Log in here.

Explore geometric properties and spatial relations that are unaffected by continuous deformations, like stretching and bending. The next time you scan a bar code on a can of soda, thank a topologist.

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

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

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

You must be logged in to see worked solutions.

Already have an account? Log in here.

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

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

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

You must be logged in to see worked solutions.

Already have an account? Log in here.

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

A. \(\{a_n\}\) is bounded.

B. \(\{a_n\}\) is monotone.

C. \(\{a_n\}\) converges.

**Definitions.**
A sequence \(\{x_n\}\) of real number is:

*bounded* if there is a real number \(M\) such that \(-M < x_n < M\) for all \(n.\)

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

*convergent* if there is a number \(L\) such that for every \(\epsilon > 0,\) there is a natural number \(N\) satisfying

\[n > N \Rightarrow |x_n - L| < \epsilon.\]

You must be logged in to see worked solutions.

Already have an account? Log in here.

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

A. \(\{a_n\}\) is bounded.

B. \(\{a_n\}\) is monotone.

C. \(\{a_n\}\) converges.

**Definitions.**
A sequence \(\{x_n\}\) of real number is:

*bounded* if there is a real number \(M\) such that \(-M < x_n < M\) for all \(n.\)

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

*convergent* if there is a number \(L\) such that for every \(\epsilon > 0,\) there is a natural number \(N\) satisfying

\[n > N \Rightarrow |x_n - L| < \epsilon.\]

You must be logged in to see worked solutions.

Already have an account? Log in here.

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 \(\{x_n\}\) of real number is:

*bounded* if there is a real number \(M\) such that \(-M < x_n < M\) for all \(n.\)

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

*convergent* if there is a number \(L\) such that for every \(\epsilon > 0,\) there is a natural number \(N\) satisfying

\[n > N \Rightarrow |x_n - L| < \epsilon.\]

You must be logged in to see worked solutions.

Already have an account? Log in here.

×

Problem Loading...

Note Loading...

Set Loading...