exercise:F87b221fb0: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow}...") |
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\newcommand{\conj}[1]{\overline{#1}} | \newcommand{\conj}[1]{\overline{#1}} | ||
\newcommand{\mathds}{\mathbb} | \newcommand{\mathds}{\mathbb} | ||
\newcommand{\dilemma}[2] { | |||
\left\{ | |||
\begin{array}{ll} | |||
#1 \\ | |||
#2 | |||
\end{array} | |||
\right. | |||
} | |||
\newcommand{\trilemma}[3] { | |||
\left\{ | |||
\begin{array}{ll} | |||
#1 \\ | |||
#2 \\ | |||
#3 | |||
\end{array} | |||
\right. | |||
} | |||
\newcommand{\cond}[1] { | |||
\begin{array}{ll} | |||
#1 | |||
\end{array} | |||
} | |||
</math></div> | </math></div> | ||
Determine whether or not each of the following | Determine whether or not each of the following sequences <math>\{ s_n \}</math> converges, and, if it does, | ||
sequences <math>\{ s_n \}</math> converges, and, if it does, | |||
evaluate the limit. | evaluate the limit. | ||
<ul style{{=}}"list-style-type:lower-alpha"><li><math>\cond{s_n = (-1)^n, & n=1,2,\ldots.}</math></li> | <ul style{{=}}"list-style-type:lower-alpha"><li><math>\cond{s_n = (-1)^n, & n=1,2,\ldots.}</math></li> | ||
<li><math>s_n = \dilemma{1+\frac1n, & \mbox{for every integer | <li><math>s_n = \dilemma{1+\frac1n, & \mbox{for every integer | ||
$n$ such that $1 \leq n \leq 10$,}} | |||
{1, & \mbox{for every integer | {1, & \mbox{for every integer $n > 10$.}}</math></li> | ||
<li><math>s_n = \dilemma{1+\frac1n, & \mbox{if | <li><math>s_n = \dilemma{1+\frac1n, & \mbox{if $n$is | ||
a positive even integer,}} | a positive even integer,}} | ||
{1, & \mbox{if | {1, & \mbox{if $n$ is a positive odd integer.}}</math></li> | ||
<li><math>s_n = \dilemma{1+\frac1n, & \mbox{for every | <li><math>s_n = \dilemma{1+\frac1n, & \mbox{for every | ||
integer | integer $n$ such that $1 \leq n \leq 10 $,}} | ||
{2, & \mbox{for every integer | {2, & \mbox{for every integer $n > 10$.}}</math></li> | ||
</ul> | </ul> | ||
Latest revision as of 03:00, 25 November 2024
[math]
\newcommand{\ex}[1]{\item }
\newcommand{\sx}{\item}
\newcommand{\x}{\sx}
\newcommand{\sxlab}[1]{}
\newcommand{\xlab}{\sxlab}
\newcommand{\prov}[1] {\quad #1}
\newcommand{\provx}[1] {\quad \mbox{#1}}
\newcommand{\intext}[1]{\quad \mbox{#1} \quad}
\newcommand{\R}{\mathrm{\bf R}}
\newcommand{\Q}{\mathrm{\bf Q}}
\newcommand{\Z}{\mathrm{\bf Z}}
\newcommand{\C}{\mathrm{\bf C}}
\newcommand{\dt}{\textbf}
\newcommand{\goesto}{\rightarrow}
\newcommand{\ddxof}[1]{\frac{d #1}{d x}}
\newcommand{\ddx}{\frac{d}{dx}}
\newcommand{\ddt}{\frac{d}{dt}}
\newcommand{\dydx}{\ddxof y}
\newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}}
\newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}}
\newcommand{\dist}{\mathrm{distance}}
\newcommand{\arccot}{\mathrm{arccot\:}}
\newcommand{\arccsc}{\mathrm{arccsc\:}}
\newcommand{\arcsec}{\mathrm{arcsec\:}}
\newcommand{\arctanh}{\mathrm{arctanh\:}}
\newcommand{\arcsinh}{\mathrm{arcsinh\:}}
\newcommand{\arccosh}{\mathrm{arccosh\:}}
\newcommand{\sech}{\mathrm{sech\:}}
\newcommand{\csch}{\mathrm{csch\:}}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\mathds}{\mathbb}
\newcommand{\dilemma}[2] {
\left\{
\begin{array}{ll}
#1 \\
#2
\end{array}
\right.
}
\newcommand{\trilemma}[3] {
\left\{
\begin{array}{ll}
#1 \\
#2 \\
#3
\end{array}
\right.
}
\newcommand{\cond}[1] {
\begin{array}{ll}
#1
\end{array}
}
[/math]
Determine whether or not each of the following sequences [math]\{ s_n \}[/math] converges, and, if it does, evaluate the limit.
- [math]\cond{s_n = (-1)^n, & n=1,2,\ldots.}[/math]
- [math]s_n = \dilemma{1+\frac1n, & \mbox{for every integer $n$ such that $1 \leq n \leq 10$,}} {1, & \mbox{for every integer $n \gt 10$.}}[/math]
- [math]s_n = \dilemma{1+\frac1n, & \mbox{if $n$is a positive even integer,}} {1, & \mbox{if $n$ is a positive odd integer.}}[/math]
- [math]s_n = \dilemma{1+\frac1n, & \mbox{for every integer $n$ such that $1 \leq n \leq 10 $,}} {2, & \mbox{for every integer $n \gt 10$.}}[/math]