How to align equations under section name, not numbering?

Question

I tried aligning equations under section names, but instead they got aligned under section numberings.
documentclass{article}
usepackage[utf8]{inputenc}
usepackage{amsmath}

title{Integration I}
author{LongKit}
date{December 5, 2020}

begin{document}

maketitle
renewcommand{thesection}{Roman{section}}
setlengthparindent{0pt}

section{The Euler's Number textit{e}}

vspace{baselineskip}

$begin{aligned}
e &= lim_{x to infty} left( 1 + frac{1}{x} right) ^{x} 
 &= sum_{x=0}^{infty} frac{1}{x!} 
e^{x} &= lim_{n to infty} left( 1 + frac{x}{n} right) ^{n} 
 &= sum_{x=0}^{infty} frac{x^n}{n!} 
end{aligned}$

end{document}

Upper: What I get; Lower: What I want
Much Appreciated!

Mico · Accepted Answer

The "twist" in your setup is that you use roman rather than arabic numerals for the section numbers. This requires setting a separate indentation amount each time section is run. The following code achieves this by creating a macro called setmylength, which sets the value of a length variable called mylength.

documentclass{article}
usepackage[T1]{fontenc}
usepackage{amsmath}

renewcommand{thesection}{Roman{section}}
setlengthparindent{0pt}

newlengthmylength
newcommandsetmylength{settowidthmylength{Largebfseriesthesectionquad}}

begin{document}

section{Euler's Number textit{e}}
setmylength

hspace{mylength}%
$begin{aligned}
e &= lim_{x to infty} Bigl( 1 + frac{1}{x} Bigr) ^{x} 
 &= sum_{x=0}^{infty} frac{1}{x!} 
e^{x} &= lim_{n to infty} Bigl( 1 + frac{x}{n} Bigr) ^{n} 
 &= sum_{x=0}^{infty} frac{x^n}{n!} 
end{aligned}$

setcounter{section}{27} % just for this example
section{Euler's Number textit{e}}
setmylength

hspace{mylength}%
$begin{aligned}
e &= lim_{x to infty} Bigl( 1 + frac{1}{x} Bigr) ^{x} 
 &= sum_{x=0}^{infty} frac{1}{x!} 
e^{x} &= lim_{n to infty} Bigl( 1 + frac{x}{n} Bigr) ^{n} 
 &= sum_{x=0}^{infty} frac{x^n}{n!} 
end{aligned}$

end{document}

Addendum to address the OP's follow-up query: If more than one alignment point is needed, replaced the aligned environment with an alginedat environment. E.g.,
$begin{alignedat}{2}
&e &&= lim_{x to infty} Bigl( 1 + frac{1}{x} Bigr) ^{x} 
&  &&= sum_{x=0}^{infty} frac{1}{x!} 
&e^{x} &&= lim_{n to infty} Bigl( 1 + frac{x}{n} Bigr) ^{n} 
&  &&= sum_{x=0}^{infty} frac{x^n}{n!} 
end{alignedat}$

Sebastiano · Answer

Here I have given an answer using nccmath, and scrextend packages with the enviroment
begin{addmargin}.... end{addmargin}

I had worked before without your MWE and I think that there are many solutions. With
begin{addmargin}[.5em]{1em}&<------
you can change the parameters of the left margin of the only equations.

documentclass[a4paper,12pt]{article}
usepackage{amsmath,amssymb}
usepackage{scrextend}
usepackage{nccmath}
begin{document}
section{The Euler's Number $e$.}
begin{addmargin}[.5em]{1em}
begin{fleqn}[parindent]
begin{align*}
 e &= lim_{x to infty} left( 1 + frac{1}{x} right) ^{x} 
 &= sum_{x=0}^{infty} frac{1}{x!} 
e^{x} &= lim_{n to infty} left( 1 + frac{x}{n} right) ^{n} 
 &= sum_{x=0}^{infty} frac{x^n}{n!} 
 end{align*}
end{fleqn} 
end{addmargin}

end{document}

I put another example, with begin{addmargin}[1em]{1em}.
documentclass[a4paper,12pt]{article}
usepackage{amsmath,amssymb}
usepackage{scrextend}
usepackage{nccmath}
begin{document}
section{The Euler's Number $e$.}
begin{addmargin}[1em]{1em}
begin{fleqn}[parindent]
begin{align*}
 e &= lim_{x to infty} left( 1 + frac{1}{x} right) ^{x} 
 &= sum_{x=0}^{infty} frac{1}{x!} 
e^{x} &= lim_{n to infty} left( 1 + frac{x}{n} right) ^{n} 
 &= sum_{x=0}^{infty} frac{x^n}{n!} 
 end{align*}
end{fleqn} 
end{addmargin}

end{document}

How to align equations under section name, not numbering?

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