1
\begin{align*}
 |S_n(x)-S_n(y)| &= |\Sum_{k=0}^n a_k(x)^{n} -\Sum_{k=0}^n a_k(y)^{n}| \\
 &= |\Sum_{k=0}^n a_k(x^{n}*y^{n})| \\
&\le \Sum_{k=0}^n$ $|a_k(x^{n}*y^{n})| \\
&=  \Sum_{k=0}^n |a_k(x-y)(x^{n-1}+x^{n-2}*y + ...+ x*y^{n-2}+ y^{n-1})| \\
&\le \Sum_{k=0}^n |a_k(x-y)(L^{n-1}+L^{n-2}*L + ...+ L*L^{n-2}+ L^{n-1})| \\
&= \Sum_{k=0}^n |a_k(x-y)(n*L^{n-1})| \\
&< \Sum_{k=0}^n |a_k*\delta*(n*L^{n-1})| \\
&\le \delta\Sum_{k=0}^n |a_k||(n*L^{n-1})| \\
\end{align*} 
5

You have two mistakes in your code:

  • \sum starts with a lower-case "s"
  • there is a pair of dollar signs $ $ that makes an error.

Other things to improve your code (thanks to the guys in the comments):

  • use \dots (probably standard, the dots are then centered) or \ldots (if you prefer baseline dots) instead of typing "..."
  • replace the "*" by \cdot, or even better leave out most of them
  • you might want to enlarge the enclosing "|" pairs. It did that here with \Bigg and \big commands, see here for more details
  • you don't need a line-break \\ at the end of the last line
  • the align* environment works for the example. But you should also consider the split environment. See for example here for details.

Full example:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{equation*}\begin{split}
    |S_n(x)-S_n(y)| &= \Bigg|\sum_{k=0}^n a_k(x)^{n} -\sum_{k=0}^n a_k(y)^{n}\Bigg| \\
    &= \Bigg|\sum_{k=0}^n a_k(x^{n}\cdot y^{n})\Bigg| \\
    &\le \sum_{k=0}^n \big|a_k(x^{n}\cdot y^{n})\big| \\
    &=  \sum_{k=0}^n \big|a_k(x-y)\big(x^{n-1}+x^{n-2}\cdot y + \dots + x\cdot y^{n-2}+ y^{n-1}\big)\big| \\
    &\le \sum_{k=0}^n \big|a_k(x-y)\big(L^{n-1}+L^{n-2}\cdot L + \dots + L\cdot L^{n-2}+ L^{n-1}\big)\big| \\
    &= \sum_{k=0}^n \big|a_k(x-y)\big(n\cdot L^{n-1}\big)\big| \\
    &< \sum_{k=0}^n \big|a_k\cdot \delta\cdot \big(n\cdot L^{n-1}\big)\big| \\
    &\le \delta\cdot\sum_{k=0}^n |a_k|\cdot\big|\big(n\cdot L^{n-1}\big)\big|
\end{split}\end{equation*}

\end{document}

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  • + \ldots + is wrong: use + \dots + instead. – GuM May 15 '17 at 20:55
  • Why is \ldots wrong? – Tiuri May 15 '17 at 20:56
  • The dots between two + signs should be centered. In any case, it’s almost always best to use \dots and let the amsmath package decide how to position the dots. – GuM May 15 '17 at 21:01
  • 1
    I like my dots to be on the baseline. According to a quick search on Wikipedia, this is perfectly fine in German and Russian, but non-standard in English. Anyway, my answer already points out that \dotsis a valid choice for this case. – Tiuri May 15 '17 at 21:05
  • 1
    Fair enough: perhaps the terseness of my comment was a bit rude, and I apologize for that. Nonetheless, I’d maintain the point that, in general, it’s best to recommend using \dots. Allow me to draw your attention to a few other slips that I’ve just noticed in your answer: (1) there should be no \\  after the last line of the align* environment; (2) perhaps (I say perhaps) it would be better to use split here; (3) vertical bars should indeed made larger in some places; (4) in the last line, I really think that \delta\cdot\sum... is better. – GuM May 15 '17 at 21:15

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