deletions | additions       diff --git a/ratio.tex b/ratio.tex  index b192a50..60b4fe2 100644  --- a/ratio.tex  +++ b/ratio.tex 
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The series converges for all real numbers so it has an infinite radius of convergence.  Determine the radius of convergence of:  \begin{align}  \sum^{\infty}_{n=0} \frac{(x\:-\:4)^n}{5^n}  \end{align}  Apply Ratio Test:  \begin{align}  \lim_{n\to\infty}\left|\:a(n\:+\:1)\hspace{1em}\div\hspace{1em}a(n)\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(x\:-\:4)^{(n\:+\:1)}}{5^{(n\:+\:1)}}\hspace{1em}\div\hspace{1em}\frac{(x\:-\:4)^n}{5^n}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(x\:-\:4)^{(n\:+\:1)}}{5^{(n\:+\:1)}}\hspace{1em}\cdot \hspace{1em}\frac{5^n}{(x\:-\:4)^n}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(x\:-\:4)^{(n\:+\:1)}}{5^{(n\:+\:1)}}\hspace{1em}\cdot \hspace{1em}\frac{5^n}{(x\:-\:4)^n}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(x\:-\:4)^{(n\:+\:1)}}{1}\hspace{1em}\cdot \hspace{1em}\frac{5^n \cdot 5^{-(n\:+\:1)}}{(x\:-\:4)^n}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(x\:-\:4)^{(n\:+\:1)}}{1}\hspace{1em}\cdot \hspace{1em}\frac{5^n \cdot 5^{(-n\:-\:1)}}{(x\:-\:4)^n}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(x\:-\:4)^{(n\:+\:1)}}{1}\hspace{1em}\cdot \hspace{1em}\frac{5^{(n\:-\:n\:-\:1)}}{(x\:-\:4)^n}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(x\:-\:4)^{(n\:+\:1)}}{1}\hspace{1em}\cdot \hspace{1em}\frac{5^{(-1)}}{(x\:-\:4)^n}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \lim_{n\to\infty}\left|\:\frac{(x\:-\:4)^{(n\:+\:1)}}{1}\hspace{1em}\cdot \hspace{1em}\frac{1}{5(x\:-\:4)^n}\:\right|\hspace{1em}<\hspace{1em} 1 \\[1em]  \end{align}  \end{document}