8c404a…: “Throughout a wonderful set”

$\newenvironment {prompt}{}{} \newcommand {\accordion }[0]{} \newcommand {\ungraded }[0]{} \newcommand {\xmDefaultPreamble }[0]{xmPreamble.tex} \newcommand {\xmDefaultPrintstyle }[0]{xmPrintstyle.sty} \newcommand {\todo }[0]{} \newcommand {\oiint }[0]{{\large \bigcirc }\kern -1.56em\iint } \newcommand {\mooculus }[0]{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}} \newcommand {\tkzTabSlope }[1]{\foreach \x /\y /\z in {#1}{\node [left = 3pt] at (Z\x 1) {\scriptsize $\y $}; \node [right = 3pt] at (Z\x 1) {\scriptsize $\z $}; }} \newcommand {\RR }[0]{\mathbb R} \newcommand {\R }[0]{\mathbb R} \newcommand {\N }[0]{\mathbb N} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\sagemath }[0]{\textsf {SageMath}} \newcommand {\d }[0]{\,d} \newcommand {\l }[0]{\ell } \newcommand {\ddx }[0]{\frac {d}{\d x}} \newcommand {\zeroOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }[0]{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }[0]{\ensuremath {\small \boldsymbol {\infty - \infty }}} \newcommand {\oneToInfty }[0]{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }[0]{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }[0]{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }[0]{\textbf } \newcommand {\unit }[0]{\mathop {}\!\mathrm } \newcommand {\eval }[1]{\bigg [ #1 \bigg ]} \newcommand {\seq }[1]{\left ( #1 \right )} \newcommand {\epsilon }[0]{\varepsilon } \newcommand {\phi }[0]{\varphi } \newcommand {\iff }[0]{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\arrowvec }[1]{{\overset {\rightharpoonup }{#1}}} \newcommand {\vec }[1]{{\overset {\boldsymbol {\rightharpoonup }}{\mathbf {#1}}}\hspace {0in}} \newcommand {\point }[1]{\left (#1\right )} \newcommand {\pt }[1]{\mathbf {#1}} \newcommand {\Lim }[2]{\lim _{\point {#1} \to \point {#2}}} \DeclareMathOperator {\proj }{\mathbf {proj}} \newcommand {\veci }[0]{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }[0]{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }[0]{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }[0]{\vec {\boldsymbol {\l }}} \newcommand {\uvec }[1]{\mathbf {\hat {#1}}} \newcommand {\utan }[0]{\mathbf {\hat {t}}} \newcommand {\unormal }[0]{\mathbf {\hat {n}}} \newcommand {\ubinormal }[0]{\mathbf {\hat {b}}} \newcommand {\dotp }[0]{\bullet } \newcommand {\cross }[0]{\boldsymbol \times } \newcommand {\grad }[0]{\boldsymbol \nabla } \newcommand {\divergence }[0]{\grad \dotp } \newcommand {\curl }[0]{\grad \cross } \newcommand {\lto }[0]{\mathop {\longrightarrow \,}\limits } \newcommand {\bar }[0]{\overline } \newcommand {\surfaceColor }[0]{violet} \newcommand {\surfaceColorTwo }[0]{redyellow} \newcommand {\sliceColor }[0]{greenyellow} \newcommand {\vector }[1]{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }[0]{} \newcommand {\luastring }[1]{"\luatexluaescapestring {#1}"} \newcommand {\luastringO }[1]{\luastring {\unexpanded \expandafter {#1}}} \newcommand {\luastringN }[1]{\luastring {\unexpanded {#1}}} \newcommand {\RestoreMathJaxEnvironment }[1]{\expandafter \let \csname #1\expandafter \endcsname \csname mathjax-#1\endcsname \expandafter \let \csname end#1\expandafter \endcsname \csname mathjax-end#1\endcsname } \newcommand {\DeclareMicrotypeVariants }[1]{} \newcommand {\DeclareMicrotypeAlias }[2]{} \newcommand {\LoadMicrotypeFile }[1]{} \newcommand {\DeclareMicrotypeFilePrefix }[1]{} \newcommand {\DeclareMicrotypeBabelHook }[2]{} \newcommand {\microtypesetup }[1]{} \newcommand {\microtypecontext }[1]{} \newcommand {\textmicrotypecontext }[2]{#2} \newcommand {\leftprotrusion }[1]{#1} \newcommand {\rightprotrusion }[1]{#1} \newcommand {\noprotrusionifhmode }[0]{} \newcommand {\lsstyle }[0]{} \newcommand {\lslig }[1]{#1} \newcommand {\maketitle }[0]{} \newcommand {\problem }[0]{ } \newcommand {\exercise }[0]{ } \newcommand {\question }[0]{ } \newcommand {\exploration }[0]{ } \newcommand {\hint }[0]{ } \newcommand {\ConfigureTheoremEnv }[1]{\renewenvironment {#1}[1][]{\refstepcounter {problem}\ifthenelse {\equal {###1}{}}{}{\HCode {<span class="theorem-like-title">}###1\HCode {</span>}}}{} \ConfigureEnv {#1}{\stepcounter {identification}\ifvmode \IgnorePar \fi \EndP \HCode {<div class="theorem-like problem-environment #1" id="problem\arabic {identification}" titletext=" \GetTranslation {#1}">}}{\ifvmode \IgnorePar \fi \EndP \HCode {</div>}\IgnoreIndent }{}{}} \newcommand {\dialogue }[0]{\begin {description}} \newcommand {\foldable }[0]{\stepcounter {identification}\ifvmode \IgnorePar \fi \EndP \HCode {<div id="foldable\arabic {identification}" class="foldable">}} \newcommand {\expandable }[0]{\stepcounter {identification}\ifvmode \IgnorePar \fi \EndP \HCode {<div data-original="expandable" id="foldable\arabic {identification}" class="foldable">}} \newcommand {\unfoldable }[1]{\HCode {<span class="unfoldable">}#1\HCode {</span>}} \newcommand {\selectAll }[0]{\refstepcounter {problem}} \newcommand {\freeResponse }[0]{\refstepcounter {problem}} \newcommand {\javascript }[0]{\NoFonts } \newcommand {\sageCell }[0]{\NoFonts } \newcommand {\sageOutput }[0]{\NoFonts } \newcommand {\sagesilent }[0]{\NoFonts } \newcommand {\ungraded }[0]{\ifvmode \IgnorePar \fi \EndP \HCode {<div class="ungraded">}\IgnoreIndent } \newcommand {\accordion }[0]{\ifvmode \IgnorePar \fi \EndP \HCode {<div class="accordion">} } \newcommand {\paragraph }[1]{\HCode {<span class="paragraphHead">}#1\HCode {</span>}\par \IgnorePar } \newcommand {\subparagraph }[1]{\HCode {<span class="subparagraphHead">}#1\HCode {</span>}\par \IgnorePar } \newcommand {\outcome }[1]{\HCode {<span class="learning-outcome">#1</span>} } \newcommand {\javascriptCode }[0]{\NoFonts } \newcommand {\GetTranslation }[1]{#1} \newcommand {\d }[0]{\,d} \newcommand {\ref }[1]{\HCode {<a class="reference" href="\##1">#1</a>}}$

Consider the power series $\sum _{k=0}^{\infty } \frac {k!}{7^k (k+1)}(x-5)^k$.

The radius of convergence for the power series is $\answer {0}$.

The lefthand endpoint of the “interval" of convergence is $x=\answer {5}$ and the righthand endpoint is $x= \answer {5}$.

This means that the power series:

does not convergence for any real $x$-values. converges only at the center $x=5$; the domain of the function is a single point rather than an interval!