Section 8.2 Sequence Properties and Limits (SQ2)
Learning Outcomes
Determine if a sequence is convergent, divergent, monotonic, or bounded, and compute limits of convergent sequences.
Activity 8.2.1.
We will consider the function \(f(x) = \displaystyle \frac{4x+8}{x}\text{.}\)
(a)
Compute the limit \(\displaystyle \lim_{x\to \infty} \frac{4x+8}{x}\text{.}\)
\(\displaystyle 0\)
\(\displaystyle 8\)
\(\displaystyle 1\)
\(\displaystyle 4\)
(b)
Determine on which intervals \(f(x)\) is increasing and/or decreasing. (Hint: compute \(f'(x)\) first.)
(c)
Which statement best describes \(f(x)\) for \(x>0\text{?}\)
\(f(x)\) is bounded above by 4
\(f(x)\) is bounded below by 4
\(f(x)\) is bounded above and below by 4
\(f(x)\) is not bounded above
\(f(x)\) is not bounded below
Definition 8.2.2.
Given a sequence \(\{x_n\}\text{:}\)
\(\{x_n\}\) is monotonically increasing if \(x_{n+1}>x_n\) for every choice of \(n\text{.}\)
\(\{x_n\}\) is monotonically non-decreasing if \(x_{n+1}\geq x_n\) for every choice of \(n\text{.}\)
\(\{x_n\}\) is monotonically decreasing if \(x_{n+1} < x_n\) for every choice of \(n\text{.}\)
\(\{x_n\}\) is monotonically non-increasing if \(x_{n+1}\leq x_n\) for every choice of \(n\text{.}\)
All of these sequences would be monotonic.
Activity 8.2.3.
Consider the sequence \(\left\{\displaystyle \frac{(-1)^n}{n}\right\}_{n=1}^\infty.\)
(a)
Compute \(x_{n+1}-x_n\text{.}\)
(b)
Which of the following is true about \(x_{n+1}-x_n\text{?}\) There can be more or less than one answer.
\(x_{n+1}-x_n> 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n\geq 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n < 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n\leq 0\) for every choice of \(n\text{.}\)
(c)
Which of the following (if any) describe \(\left\{\displaystyle \frac{(-1)^n}{n}\right\}_{n=1}^\infty?\)
Monotonically increasing.
Monotonically non-decreasing.
Monotonically decreasing.
Monotonically non-increasing.
Activity 8.2.4.
Consider the sequence \(\left\{\displaystyle \frac{n^2+1}{n}\right\}_{n=1}^\infty.\)
(a)
Compute \(x_{n+1}-x_n\text{.}\)
(b)
Which of the following is true about \(x_{n+1}-x_n\text{?}\) There can be more or less than one answer.
\(x_{n+1}-x_n> 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n\geq 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n < 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n\leq 0\) for every choice of \(n\text{.}\)
(c)
Which of the following (if any) describe \(\left\{\displaystyle \frac{n^2+1}{n}\right\}_{n=1}^\infty?\)
Monotonically increasing.
Monotonically non-decreasing.
Monotonically decreasing.
Monotonically non-increasing.
Activity 8.2.5.
Consider the sequence \(\left\{\displaystyle \frac{n+1}{n}\right\}_{n=1}^\infty.\)
(a)
Compute \(x_{n+1}-x_n\text{.}\)
(b)
Which of the following is true about \(x_{n+1}-x_n\text{?}\) There can be more or less than one answer.
\(x_{n+1}-x_n> 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n\geq 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n < 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n\leq 0\) for every choice of \(n\text{.}\)
(c)
Which of the following (if any) describe \(\left\{\displaystyle \frac{n+1}{n}\right\}_{n=1}^\infty?\)
Monotonically increasing.
Monotonically non-decreasing.
Monotonically decreasing.
Monotonically non-increasing.
Activity 8.2.6.
Consider the sequence \(\left\{\displaystyle \frac{2}{3^n}\right\}_{n=0}^\infty.\)
(a)
Compute \(x_{n+1}-x_n\text{.}\)
(b)
Which of the following is true about \(x_{n+1}-x_n\text{?}\) There can be more or less than one answer.
\(x_{n+1}-x_n> 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n\geq 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n < 0\) for every choice of \(n\text{.}\)
\(x_{n+1}-x_n\leq 0\) for every choice of \(n\text{.}\)
(c)
Which of the following (if any) describe \(\left\{\displaystyle \frac{2}{3^n}\right\}_{n=0}^\infty?\)
Monotonically increasing.
Monotonically non-decreasing.
Monotonically decreasing.
Monotonically non-increasing.
Definition 8.2.7.
A sequence \(\{x_n\}\) is bounded if there are real numbers \(b_u, b_{\ell}\) such that
for every \(n\text{.}\)
Activity 8.2.8.
Consider the sequence \(\left\{\displaystyle \frac{(-1)^n}{n}\right\}_{n=1}^\infty\) from Activity 8.2.3.
(a)
Is there a \(b_u\) such that \(x_n\leq b_u\) for every \(n\text{?}\) If so, what would be one such \(b_u\text{?}\)
(b)
Is there a \(b_\ell\) such that \(b_\ell \leq x_n\) for every \(n\text{?}\) If so, what would be one such \(b_\ell\text{?}\)
(c)
Is \(\left\{\displaystyle \frac{(-1)^n}{n}\right\}_{n=1}^\infty\) bounded?
Activity 8.2.9.
Consider the sequence \(\left\{\displaystyle \frac{n^2+1}{n}\right\}_{n=1}^\infty\) from Activity 8.2.4.
(a)
Is there a \(b_u\) such that \(x_n\leq b_u\) for every \(n\text{?}\) If so, what would be one such \(b_u\text{?}\)
(b)
Is there a \(b_\ell\) such that \(b_\ell \leq x_n\) for every \(n\text{?}\) If so, what would be one such \(b_\ell\text{?}\)
(c)
Is \(\left\{\displaystyle \frac{n^2+1}{n}\right\}_{n=1}^\infty\) bounded?
Activity 8.2.10.
Consider the sequence \(\left\{\displaystyle \frac{n+1}{n}\right\}_{n=1}^\infty\) from Activity 8.2.5.
(a)
Is there a \(b_u\) such that \(x_n\leq b_u\) for every \(n\text{?}\) If so, what would be one such \(b_u\text{?}\)
(b)
Is there a \(b_\ell\) such that \(b_\ell \leq x_n\) for every \(n\text{?}\) If so, what would be one such \(b_\ell\text{?}\)
(c)
Is \(\left\{\displaystyle \frac{n+1}{n}\right\}_{n=1}^\infty\) bounded?
Activity 8.2.11.
Consider the sequence \(\left\{\displaystyle \frac{2}{3^n}\right\}_{n=1}^\infty\) from Activity 8.2.6.
(a)
Is there a \(b_u\) such that \(x_n\leq b_u\) for every \(n\text{?}\) If so, what would be one such \(b_u\text{?}\)
(b)
Is there a \(b_\ell\) such that \(b_\ell \leq x_n\) for every \(n\text{?}\) If so, what would be one such \(b_\ell\text{?}\)
(c)
Is \(\left\{\displaystyle \frac{2}{3^n}\right\}_{n=1}^\infty\) bounded?
Definition 8.2.12.
Given a sequence \(\{x_n\}\text{,}\) we say \(x_n\) has limit \(L\text{,}\) denoted
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 \(L\text{.}\) If no such \(L\) exists, we say \(\{x_n\}\) does not converge.
Activity 8.2.13.
(a)
For each of the following, determine if the sequence converges.
\(\displaystyle \left\{\displaystyle \frac{(-1)^n}{n}\right\}_{n=1}^\infty.\)
\(\displaystyle \left\{\displaystyle \frac{n^2+1}{n}\right\}_{n=1}^\infty.\)
\(\displaystyle \left\{\displaystyle \frac{n+1}{n}\right\}_{n=1}^\infty.\)
\(\displaystyle \left\{\displaystyle \frac{2}{3^n}\right\}_{n=0}^\infty.\)
(b)
Where possible, find the limit of the sequence.
Activity 8.2.14.
(a)
Determine to what value \(\left\{\displaystyle \frac{4n}{n+1}\right\}_{n=0}^\infty\) converges.
(b)
Which of the following ia most likely true about \(\left\{\displaystyle \frac{4n(-1)^n}{n+1}\right\}_{n=0}^\infty\text{?}\)
\(\left\{\displaystyle \frac{4n(-1)^n}{n+1}\right\}_{n=0}^\infty\) converges to 4.
\(\left\{\displaystyle \frac{4n(-1)^n}{n+1}\right\}_{n=0}^\infty\) converges to 0.
\(\left\{\displaystyle \frac{4n(-1)^n}{n+1}\right\}_{n=0}^\infty\) converges to -4.
\(\left\{\displaystyle \frac{4n(-1)^n}{n+1}\right\}_{n=0}^\infty\) does not converge.
Activity 8.2.15.
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. (a)
\(\left\{\displaystyle 3n\right\}_{n=0}^\infty.\)(b)
\(\left\{\displaystyle \frac{n^3}{3^n}\right\}_{n=0}^\infty.\)(c)
\(\left\{\displaystyle \frac{n}{n+3}\right\}_{n=1}^\infty.\)(d)
\(\left\{\displaystyle \frac{(-1)^n}{n+3}\right\}_{n=1}^\infty.\)
Fact 8.2.16.
If a sequence is monotonic and bounded, then it is convergent.