Learning Outcomes
- Determine if a sequence is convergent, divergent, monotonic, or bounded, and compute limits of convergent sequences.
Section 8.2 Sequence Properties and Limits (SQ2)
Subsection 8.2.1 Activities
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
\begin{equation*}
b_{\ell}\leq x_n \leq b_u
\end{equation*}
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
\begin{equation*}
\lim_{n\to\infty} x_n=L
\end{equation*}
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.
