Upper & Lower limits
Example - Week 2
Example 2.4.1.
\(\{x_n\}\) where,
x_n :=
\begin{cases}
\frac{n+1}{n} & \text{if } n \text{ is odd,} \\
0 & \text{if } n \text{ is even.}
\end{cases}
Example 2.4.1.
\(\{x_n\}\) where,
x_n :=
\begin{cases}
\frac{n+1}{n} & \text{if } n \text{ is odd,} \\
0 & \text{if } n \text{ is even.}
\end{cases}
\left\{ 2, 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, 0, \frac{16}{15}, 0, \frac{18}{17} \ldots \right\}
\(n=1\)
\(2\)
\(n=2\)
\(0\)
\(n=3\)
\(\displaystyle\frac{4}{3}\)
\(n=4\)
\(0\)
\(n=5\)
\(\displaystyle \frac65\)
Example 2.4.1.
\left\{ 2, 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, 0, \frac{16}{15}, 0, \frac{18}{17} \ldots \right\}
\(\{x_n\}\) where,
x_n :=
\begin{cases}
\frac{n+1}{n} & \text{if } n \text{ is odd,} \\
0 & \text{if } n \text{ is even.}
\end{cases}
Example 2.4.1.
\(b_n=\inf\{x_k:k\geq n\}\)
\(\displaystyle\liminf_{n\rightarrow \infty}x_n\)
\(b_1=\inf\{x_k:k\geq 1\}\)
\(b_2=\inf\{x_k:k\geq 2\}\)
\(b_3=\inf\{x_k:k\geq 3\}\)
\(=\displaystyle\lim_{n\rightarrow \infty}b_n\)
\left\{ 2, 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, 0, \frac{16}{15}, 0, \frac{18}{17} \ldots \right\}
=\inf\left\{ 2, 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, \ldots \right\}
=\inf \left\{ 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, \ldots \right\}
=\inf \left\{ \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, \ldots \right\}
\(=0\)
\(=0\)
\(=0\)
\(=0\)
\(=0\)
Example 2.4.1.
\left\{ 2, 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, 0, \frac{16}{15}, 0, \frac{18}{17} \ldots \right\}
\(\displaystyle\liminf_{n\rightarrow \infty}x_n\)
\(=\displaystyle\lim_{n\rightarrow \infty}b_n\)
\(=0\)
Example 2.4.1.
\(a_n=\sup\{x_k:k\geq n\}\)
\(\displaystyle\limsup_{n\rightarrow \infty}x_n\)
\(a_1=\sup\{x_k:k\geq 1\}\)
\(a_2=\sup\{x_k:k\geq 2\}\)
\(a_3=\sup\{x_k:k\geq 3\}\)
\(=\displaystyle\lim_{n\rightarrow \infty}a_n\)
\left\{ 2, 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, 0, \frac{16}{15}, 0, \frac{18}{17} \ldots \right\}
=\sup\left\{ 2, 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, \ldots \right\}
=\sup \left\{ 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, \ldots \right\}
=\sup \left\{ \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, \ldots \right\}
\(=2\)
\(=\dfrac43\)
\(=\dfrac43\)
\(=1\)
\(a_4=\sup\{x_k:k\geq 4\}\)
=\sup \left\{ 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, \ldots \right\}
\(=\dfrac65\)
=
\begin{cases}
\frac{n+1}{n} & \text{if } n \text{ is odd,} \\
\frac{n+2}{n+1} & \text{if } n \text{ is even.}
\end{cases}
Example 2.4.1.
\left\{ 2, 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, 0, \frac{16}{15}, 0, \frac{18}{17} \ldots \right\}
\(\displaystyle\limsup_{n\rightarrow \infty}x_n\)
\(=\displaystyle\lim_{n\rightarrow \infty}a_n\)
\(=1\)
Example 2.4.1.
\(\displaystyle\limsup_{n\rightarrow \infty}x_n=1\)
\left\{ 2, 0, \frac{4}{3}, 0, \frac{6}{5}, 0, \frac{8}{7}, 0, \frac{10}{9}, 0, \frac{12}{11}, 0, \frac{14}{13}, 0, \frac{16}{15}, 0, \frac{18}{17} \ldots \right\}
\(\displaystyle\liminf_{n\rightarrow \infty}x_n=0\)
Upper/Lower limits: Example
By Juan Carlos Ponce Campuzano
Upper/Lower limits: Example
Upper/Lower limits: Example
