$\displaystyle \sum^{\infty}_{n=1} \frac{1+5^n}{8^n}$
I think I am over thinking this. I tried using the ratio test and get $\displaystyle \frac{6}{8}$ but that isn't it. Thanks in advance!
Hello, mollymcf2009!
The ratio test goes like this . . .$\displaystyle \sum^{\infty}_{n=1} \frac{1+5^n}{8^n}$
. . $\displaystyle R \;=\;\frac{a_{n+1}}{a_n} \;=\;\frac{1+5^{n+1}}{8^{n+1}}\cdot\frac{8^n}{1 + 5^n} \;=\;\frac{8^n}{8^{n+1}}\cdot\frac{1+5^{n+1}}{1+5^ n} \;=\;\frac{1}{8}\cdot\frac{1 + 5^{n+1}}{1+5^n} $
Divide top and bottom by $\displaystyle 5^n\!:\;\;\frac{1}{8}\cdot\frac{\frac{1}{5^n} + 5}{\frac{1}{5^n} + 1} $
Then: .$\displaystyle \lim_{n\to\infty}R \;=\;\lim_{n\to\infty}\,\frac{1}{8}\cdot\frac{\fra c{1}{5^n} + 5}{\frac{1}{5^n} + 1} \;=\;\frac{1}{8}\cdot\frac{0+5}{0+1} \;=\;\frac{5}{8}$