deletions | additions       diff --git a/ratio.tex b/ratio.tex  index f9e605d..860abee 100644  --- a/ratio.tex  +++ b/ratio.tex 
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Apply Ratio Test:  \begin{align}  \lim_{n\to\infty}\left|\:a(n\:+\:1)\hspace{1em}\div\hspace{1em}a(n)\:\right| L\hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:a(n\:+\:1)\hspace{1em}\div\hspace{1em}a(n)\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{(n\:-\:2\:+\:1)}{(n\:+\:1\:+\:1)!}\hspace{1em}\div\hspace{1em}\frac{(n\:-\:2)}{(n\:+\:1)!}\:\right| \hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:\frac{(n\:-\:2\:+\:1)}{(n\:+\:1\:+\:1)!}\hspace{1em}\div\hspace{1em}\frac{(n\:-\:2)}{(n\:+\:1)!}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{(n\:-\:1)}{(n\:+\:2)!}\hspace{1em}\div\hspace{1em}\frac{(n\:-\:2)}{(n\:+\:1)!}\:\right| \hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:\frac{(n\:-\:1)}{(n\:+\:2)!}\hspace{1em}\div\hspace{1em}\frac{(n\:-\:2)}{(n\:+\:1)!}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{(n\:-\:1)}{(n\:+\:2)!}\hspace{1em}\cdot \hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:\frac{(n\:-\:1)}{(n\:+\:2)!}\hspace{1em}\cdot  \hspace{1em}\frac{(n\:+\:1)!}{(n\:-\:2)}\:\right| \\[1em] \lim_{n\to\infty}\left|\:\frac{(n\:-\:1)(n\:+\:1)!}{(n\:-\:2)(n\:+\:2)!}\:\right| \hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:\frac{(n\:-\:1)(n\:+\:1)!}{(n\:-\:2)(n\:+\:2)!}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{(n\:-\:1)(n\:+\:1)!}{(n\:-\:2)(n\:+\:2)(n\:+\:1)!}\:\right| \hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:\frac{(n\:-\:1)(n\:+\:1)!}{(n\:-\:2)(n\:+\:2)(n\:+\:1)!}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{(n\:-\:1)}{(n\:-\:2)(n\:+\:2)}\:\right| \hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:\frac{(n\:-\:1)}{(n\:-\:2)(n\:+\:2)}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{(n\:-\:1)}{(n^2\:+\:2n\:-\:2n\:-\:4)}\:\right| \hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:\frac{(n\:-\:1)}{(n^2\:+\:2n\:-\:2n\:-\:4)}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{(n\:-\:1)}{(n^2\:-\:4)}\:\right| \hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:\frac{(n\:-\:1)}{(n^2\:-\:4)}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{n\left(\frac{n}{n}\:-\:\frac{1}{n}\right)}{n^2\left(\frac{n^2}{n^2}\:-\:\frac{4}{n^2}\right)}\:\right| \hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:\frac{n\left(\frac{n}{n}\:-\:\frac{1}{n}\right)}{n^2\left(\frac{n^2}{n^2}\:-\:\frac{4}{n^2}\right)}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{\left(1\:-\:\frac{1}{n}\right)}{n\left(1\:-\:\frac{4}{n^2}\right)}\:\right| \hspace{1em}&=\hspace{1em}\lim_{n\to\infty}\left|\:\frac{\left(1\:-\:\frac{1}{n}\right)}{n\left(1\:-\:\frac{4}{n^2}\right)}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{\left(1\:-\:\frac{1}{\infty}\right)}{\infty\left(1\:-\:\frac{4}{\infty}\right)}\:\right| \hspace{1em}&=\hspace{1em}\left|\:\frac{\left(1\:-\:\frac{1}{\infty}\right)}{\infty\left(1\:-\:\frac{4}{\infty}\right)}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{\left(1\:-\:0\right)}{\infty\left(1\:-\:0\right)}\:\right| \hspace{1em}&=\hspace{1em}\left|\:\frac{\left(1\:-\:0\right)}{\infty\left(1\:-\:0\right)}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{\left(1\right)}{\infty\left(1\right)}\:\right| \hspace{1em}&=\hspace{1em}\left|\:\frac{\left(1\right)}{\infty\left(1\right)}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:\frac{1}{\infty}\:\right| \hspace{1em}&=\hspace{1em}\left|\:\frac{1}{\infty}\:\right|  \\[1em] \lim_{n\to\infty}\left|\:0\:\right| \hspace{1em}&=\hspace{1em}\left|\:0\:\right|  \\[1em] \lim_{n\to\infty}0 L\hspace{1em}&=\hspace{1em}0  \\[1em] \end{align}  \end{document}