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Topology

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.

Monotonic Sequences

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

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

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

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

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

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

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

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

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