deletions | additions       diff --git a/ratio.tex b/ratio.tex  index 23855c8..9f557ff 100644  --- a/ratio.tex  +++ b/ratio.tex 
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\lim_{n\to\infty}\left|\:\frac{x^{(n\:+\:1)}}{(n\:+\:1\:+\:1)!}\hspace{1em}\div\hspace{1em}\frac{x^n}{(n\:+\:1)!}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{x^{(n\:+\:1)}}{(n\:+\:2)!}\hspace{1em}\div\hspace{1em}\frac{x^n}{(n\:+\:1)!}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{x^{(n\:+\:1)}}{(n\:+\:2)!}\hspace{1em}\cdot \hspace{1em}\frac{(n\:+\:1)!}{x^n}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{x^{(n\:+\:1)}\cdot x^{(-n)}}{(n\:+\:2)!}\hspace{1em}\cdot \hspace{1em}\frac{(n\:+\:1)!}{1}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{x^{(n\:+\:1\:-\:n)}}{(n\:+\:2)!}\hspace{1em}\cdot \hspace{1em}\frac{(n\:+\:1)!}{1}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{x^{(1)}}{(n\:+\:2)!}\hspace{1em}\cdot \hspace{1em}\frac{(n\:+\:1)!}{1}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{x}{(n\:+\:2)!}\hspace{1em}\cdot \hspace{1em}\frac{(n\:+\:1)!}{1}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(n\:+\:1)! \cdot x}{(n\:+\:2)!}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(n\:+\:1)! \cdot x}{(n\:+\:2)(n\:+\:1)!}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{x}{(n\:+\:2)}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{x}{n\left(\frac{n}{n}\:+\:\frac{2}{n}\right)}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \end{align}  \end{document}