exercise:B672c151e2: 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. | |||
} | |||
</math></div> | </math></div> | ||
Let <math>r</math> be a real number, | |||
and consider the sequence <math>1, r, r^2, r^3, \ldots.</math> | Let <math>r</math> be a real number, and consider the sequence <math>1, r, r^2, r^3, \ldots.</math> | ||
Show that the sequence converges if and only | Show that the sequence converges if and only | ||
if <math>-1 < r \leq 1</math>, and that | if <math>-1 < r \leq 1</math>, and that | ||
| Line 40: | Line 58: | ||
\lim_{n\goesto\infty} r^n = | \lim_{n\goesto\infty} r^n = | ||
\trilemma{0 & \mbox{if $-1 < r < 1$,}} | \trilemma{0 & \mbox{if $-1 < r < 1$,}} | ||
{1 & \mbox{if | {1 & \mbox{if $r=1$,}} | ||
{\infty & \mbox{if | {\infty & \mbox{if $r > 1$.}} | ||
</math> | </math> | ||
What is the behavior of the sequence for | What is the behavior of the sequence for | ||
Latest revision as of 02:56, 25 November 2024
[math]
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#3
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[/math]
Let [math]r[/math] be a real number, and consider the sequence [math]1, r, r^2, r^3, \ldots.[/math] Show that the sequence converges if and only if [math]-1 \lt r \leq 1[/math], and that
[[math]]
\lim_{n\goesto\infty} r^n =
\trilemma{0 & \mbox{if $-1 \lt r \lt 1$,}}
{1 & \mbox{if $r=1$,}}
{\infty & \mbox{if $r \gt 1$.}}
[[/math]]
What is the behavior of the sequence for [math]r=-1[/math] and for [math]r \lt -1[/math]? (Hint: Let [math]r^n = e^{n \ln r}[/math], for [math]r \gt 0[/math].)