TBIL-Cal Sequence Properties and Limits (SQ2)

${x_{n}}$ is monotonically increasing if $x_{n + 1} > x_{n}$ for every choice of $n .$
${x_{n}}$ is monotonically non-decreasing if $x_{n + 1} \geq x_{n}$ for every choice of $n .$
${x_{n}}$ is monotonically decreasing if $x_{n + 1} < x_{n}$ for every choice of $n .$
${x_{n}}$ is monotonically non-increasing if $x_{n + 1} \leq x_{n}$ for every choice of $n .$

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All of these sequences would be monotonic.

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Activity 8.2.3.

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Consider the sequence

{\frac{(- 1)^{n}}{n}}_{n = 1}^{\infty} .

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(a)

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Compute

x_{n + 1} - x_{n} .

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(b)

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Which of the following is true about

x_{n + 1} - x_{n} ?

There can be more or less than one answer.

$x_{n + 1} - x_{n} > 0$ for every choice of $n .$
$x_{n + 1} - x_{n} \geq 0$ for every choice of $n .$
$x_{n + 1} - x_{n} < 0$ for every choice of $n .$
$x_{n + 1} - x_{n} \leq 0$ for every choice of $n .$

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(c)

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Which of the following (if any) describe

{\frac{(- 1)^{n}}{n}}_{n = 1}^{\infty} ?

Monotonically increasing.
Monotonically non-decreasing.
Monotonically decreasing.
Monotonically non-increasing.

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Activity 8.2.4.

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Consider the sequence

{\frac{n^{2} + 1}{n}}_{n = 1}^{\infty} .

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(a)

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Compute

x_{n + 1} - x_{n} .

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(b)

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Which of the following is true about

x_{n + 1} - x_{n} ?

There can be more or less than one answer.

$x_{n + 1} - x_{n} > 0$ for every choice of $n .$
$x_{n + 1} - x_{n} \geq 0$ for every choice of $n .$
$x_{n + 1} - x_{n} < 0$ for every choice of $n .$
$x_{n + 1} - x_{n} \leq 0$ for every choice of $n .$

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(c)

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Which of the following (if any) describe

{\frac{n^{2} + 1}{n}}_{n = 1}^{\infty} ?

Monotonically increasing.
Monotonically non-decreasing.
Monotonically decreasing.
Monotonically non-increasing.

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Activity 8.2.5.

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Consider the sequence

{\frac{n + 1}{n}}_{n = 1}^{\infty} .

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(a)

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Compute

x_{n + 1} - x_{n} .

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(b)

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Which of the following is true about

x_{n + 1} - x_{n} ?

There can be more or less than one answer.

$x_{n + 1} - x_{n} > 0$ for every choice of $n .$
$x_{n + 1} - x_{n} \geq 0$ for every choice of $n .$
$x_{n + 1} - x_{n} < 0$ for every choice of $n .$
$x_{n + 1} - x_{n} \leq 0$ for every choice of $n .$

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(c)

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Which of the following (if any) describe

{\frac{n + 1}{n}}_{n = 1}^{\infty} ?

Monotonically increasing.
Monotonically non-decreasing.
Monotonically decreasing.
Monotonically non-increasing.

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Activity 8.2.6.

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Consider the sequence

{\frac{2}{3^{n}}}_{n = 0}^{\infty} .

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(a)

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Compute

x_{n + 1} - x_{n} .

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(b)

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Which of the following is true about

x_{n + 1} - x_{n} ?

There can be more or less than one answer.

$x_{n + 1} - x_{n} > 0$ for every choice of $n .$
$x_{n + 1} - x_{n} \geq 0$ for every choice of $n .$
$x_{n + 1} - x_{n} < 0$ for every choice of $n .$
$x_{n + 1} - x_{n} \leq 0$ for every choice of $n .$

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(c)

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Which of the following (if any) describe

{\frac{2}{3^{n}}}_{n = 0}^{\infty} ?

Monotonically increasing.
Monotonically non-decreasing.
Monotonically decreasing.
Monotonically non-increasing.

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Definition 8.2.7.

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A sequence

{x_{n}}

is bounded if there are real numbers

b_{u}, b_{ℓ}

such that

b_{ℓ} \leq x_{n} \leq b_{u}

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for every

n .

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Activity 8.2.8.

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Consider the sequence

{\frac{(- 1)^{n}}{n}}_{n = 1}^{\infty}

from Activity 8.2.3.

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(a)

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Is there a

b_{u}

such that

x_{n} \leq b_{u}

for every

n ?

If so, what would be one such

b_{u} ?

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(b)

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Is there a

b_{ℓ}

such that

b_{ℓ} \leq x_{n}

for every

n ?

If so, what would be one such

b_{ℓ} ?

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(c)

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{\frac{(- 1)^{n}}{n}}_{n = 1}^{\infty}

bounded?

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Activity 8.2.9.

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Consider the sequence

{\frac{n^{2} + 1}{n}}_{n = 1}^{\infty}

from Activity 8.2.4.

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(a)

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Is there a

b_{u}

such that

x_{n} \leq b_{u}

for every

n ?

If so, what would be one such

b_{u} ?

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(b)

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Is there a

b_{ℓ}

such that

b_{ℓ} \leq x_{n}

for every

n ?

If so, what would be one such

b_{ℓ} ?

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(c)

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{\frac{n^{2} + 1}{n}}_{n = 1}^{\infty}

bounded?

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Activity 8.2.10.

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Consider the sequence

{\frac{n + 1}{n}}_{n = 1}^{\infty}

from Activity 8.2.5.

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(a)

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Is there a

b_{u}

such that

x_{n} \leq b_{u}

for every

n ?

If so, what would be one such

b_{u} ?

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(b)

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Is there a

b_{ℓ}

such that

b_{ℓ} \leq x_{n}

for every

n ?

If so, what would be one such

b_{ℓ} ?

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(c)

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{\frac{n + 1}{n}}_{n = 1}^{\infty}

bounded?

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Activity 8.2.11.

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Consider the sequence

{\frac{2}{3^{n}}}_{n = 1}^{\infty}

from Activity 8.2.6.

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(a)

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Is there a

b_{u}

such that

x_{n} \leq b_{u}

for every

n ?

If so, what would be one such

b_{u} ?

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(b)

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Is there a

b_{ℓ}

such that

b_{ℓ} \leq x_{n}

for every

n ?

If so, what would be one such

b_{ℓ} ?

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(c)

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{\frac{2}{3^{n}}}_{n = 1}^{\infty}

bounded?

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Definition 8.2.12.

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Given a sequence

{x_{n}},

we say

x_{n}

has limit

L,

denoted

lim_{n \to \infty} x_{n} = L

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if we can make

x_{n}

as close to

L

as we like by making

n

sufficiently large. If such an

L

exists, we say ${x_{n}}$ converges to $L .$ If no such

L

exists, we say

{x_{n}}

diverges.

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Activity 8.2.13.

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(a)

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For each of the following, determine if the sequence converges.

${\frac{(- 1)^{n}}{n}}_{n = 1}^{\infty} .$
${\frac{n^{2} + 1}{n}}_{n = 1}^{\infty} .$
${\frac{n + 1}{n}}_{n = 1}^{\infty} .$
${\frac{2}{3^{n}}}_{n = 0}^{\infty} .$

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(b)

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Where possible, find the limit of the sequence.

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Activity 8.2.14.

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(a)

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Determine to what value

{\frac{4 n}{n + 1}}_{n = 0}^{\infty}

converges.

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(b)

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Which of the following is most likely true about

{\frac{4 n (- 1)^{n}}{n + 1}}_{n = 0}^{\infty} ?

${\frac{4 n (- 1)^{n}}{n + 1}}_{n = 0}^{\infty}$ converges to 4.
${\frac{4 n (- 1)^{n}}{n + 1}}_{n = 0}^{\infty}$ converges to 0.
${\frac{4 n (- 1)^{n}}{n + 1}}_{n = 0}^{\infty}$ converges to -4.
${\frac{4 n (- 1)^{n}}{n + 1}}_{n = 0}^{\infty}$ does not converge.

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Activity 8.2.15.

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For each of the following sequences, determine which of the properties: monotonic, bounded and convergent, the sequence satisfies. If a sequence is convergent, determine to what it converges.

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