Alignment problem with large subscript in sum

If I have understood you correctly, then the sum of alphas (1 + \alpha + ...) should be moved to the left to avoid the overful \hbox warning:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
This is a really long line of text to show where the margins of the page are.
This is a really long line of text to show where the margins of the page are.
This is a really long line of text to show where the margins of the page are.
\[
  \sum_{\alpha \in \{a,b,c,d,e,f,g,h,i,j,k,\ell,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}}
  \hspace{0pt minus 12.2pt}% value taken from the overful \hbox message
  1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 + \alpha^5 + \alpha^6 + \alpha^7 + \alpha^8 + \alpha^9
\]
This is a really long line of text to show where the margins of the page are.
This is a really long line of text to show where the margins of the page are.
This is a really long line of text to show where the margins of the page are.
\[
  \sum_{\alpha \in \{a,b,c,d,e,f,g,h,i,j,k,\ell,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}}
  \hspace{0pt minus 1fil}% allows natural width for the spacing around plus operators
  1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 + \alpha^5 + \alpha^6 + \alpha^7 + \alpha^8 + \alpha^9
\]
\end{document}

With avoiding the magic number

\documentclass{article}
\usepackage{amsmath}

\newsavebox\testbox

\begin{document}
This is a really long line of text to show where the margins of the page are.
This is a really long line of text to show where the margins of the page are.
This is a really long line of text to show where the margins of the page are.
\newcommand*{\alphasum}[1]{%
  \sum_{\alpha \in \{a,b,c,d,e,f,g,h,i,j,k,\ell,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}}
  \hspace{0pt minus #1}
  1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 + \alpha^5 + \alpha^6 + \alpha^7
}
\sbox\testbox{$\displaystyle\alphasum{0pt}$}%
\[
  \ifdim\wd\testbox>\linewidth
    \alphasum{\dimexpr\wd\testbox-\linewidth}%
  \else
    \alphasum{0pt}%
  \fi
\]
This is a really long line of text to show where the margins of the page are.
This is a really long line of text to show where the margins of the page are.
This is a really long line of text to show where the margins of the page are.
\[
  \sum_{\alpha \in \{a,b,c,d,e,f,g,h,i,j,k,\ell,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}}
  \hspace{0pt minus 1fil}
  1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 + \alpha^5 + \alpha^6 + \alpha^7
\]
\end{document}

Keep it simple

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\[
  \sum_{\alpha \in \{a,b,c,\dots, z\}}
  \hspace{0pt minus 1fil}
  1 + \alpha + \alpha^2 + \alpha^3 + \dots + \alpha^9
\]
\end{document}

+1 for Heiko's answer, and in particular his advice about keeping it simple.

I would actually re-write your expression to use a local definition

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\[
  \sum_{\alpha \in S}
  1 + \alpha + \alpha^2 + \alpha^3 + \dots + \alpha^9
\]
where
\[
S=\{a,b,c,\dots, z\}
\]
\end{document}