this is for holding javascript data
redthumb edited ratio.tex
over 9 years ago
Commit id: 0738ea25a7183d91fef93fa7d0601a76aa0e275f
deletions | additions
diff --git a/ratio.tex b/ratio.tex
index 75095e0..08ba45b 100644
--- a/ratio.tex
+++ b/ratio.tex
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\lim_{n\to\infty}\left|\:\frac{(n\:+\:1)\cdot x^{(n\:+\:1)}}{2\cdot3^{(n\:+\:1)}}\hspace{1em}\div\hspace{1em}\frac{(n\:+\:1)\cdot x^n}{2\cdot3^{n}}\:\right|\hspace{1em}<\hspace{1em}1 \\[1em]
\lim_{n\to\infty}\left|\:\frac{(n\:+\:1)\cdot x^{(n\:+\:1)}}{2\cdot3^{(n\:+\:1)}}\hspace{1em}\cdot\hspace{1em}\frac{2\cdot3^{n}}{(n\:+\:1)\cdot x^n}\:\right|\hspace{1em}<\hspace{1em}1 \\[1em]
\lim_{n\to\infty}\left|\:\frac{(n\:+\:1)\cdot x^{(n\:+\:1)}\cdot x^{(-n)}}{2\cdot3^{(n\:+\:1)}}\hspace{1em}\cdot\hspace{1em}\frac{2\cdot3^{n}}{(n\:+\:1)}\:\right|\hspace{1em}<\hspace{1em}1 \\[1em]
\lim_{n\to\infty}\left|\:\frac{(n\:+\:1)\cdot x^{(n\:+\:1\:-\:n)}}{2\cdot3^{(n\:+\:1)}}\hspace{1em}\cdot\hspace{1em}\frac{2\cdot3^{n}}{(n\:+\:1)}\:\right|\hspace{1em}<\hspace{1em}1 \\[1em]
\end{align}