# Thread: [SOLVED] Infinite series #3

1. ## [SOLVED] Infinite series #3

$\sum^{\infty}_{n=1} \frac{1+5^n}{8^n}$

I think I am over thinking this. I tried using the ratio test and get $\frac{6}{8}$ but that isn't it. Thanks in advance!

2. $\sum \frac{1+5^n}{8^n} = \sum \left( \frac{1}{8^n} + \frac{5^n}{8^n}\right) = \sum \left( \frac{1}{8}\right)^n + \sum \left(\frac{5}{8}\right)^n$

Are the two right hand side series convergent?

3. Hello, mollymcf2009!

$\sum^{\infty}_{n=1} \frac{1+5^n}{8^n}$
The ratio test goes like this . . .

. . $R \;=\;\frac{a_{n+1}}{a_n} \;=\;\frac{1+5^{n+1}}{8^{n+1}}\cdot\frac{8^n}{1 + 5^n} \;=\;\frac{8^n}{8^{n+1}}\cdot\frac{1+5^{n+1}}{1+5^ n} \;=\;\frac{1}{8}\cdot\frac{1 + 5^{n+1}}{1+5^n}$

Divide top and bottom by $5^n\!:\;\;\frac{1}{8}\cdot\frac{\frac{1}{5^n} + 5}{\frac{1}{5^n} + 1}$

Then: . $\lim_{n\to\infty}R \;=\;\lim_{n\to\infty}\,\frac{1}{8}\cdot\frac{\fra c{1}{5^n} + 5}{\frac{1}{5^n} + 1} \;=\;\frac{1}{8}\cdot\frac{0+5}{0+1} \;=\;\frac{5}{8}$

4. Originally Posted by Soroban
Hello, mollymcf2009!

The ratio test goes like this . . .

. . $R \;=\;\frac{a_{n+1}}{a_n} \;=\;\frac{1+5^{n+1}}{8^{n+1}}\cdot\frac{8^n}{1 + 5^n} \;=\;\frac{8^n}{8^{n+1}}\cdot\frac{1+5^{n+1}}{1+5^ n} \;=\;\frac{1}{8}\cdot\frac{1 + 5^{n+1}}{1+5^n}$

Divide top and bottom by $5^n\!:\;\;\frac{1}{8}\cdot\frac{\frac{1}{5^n} + 5}{\frac{1}{5^n} + 1}$

Then: . $\lim_{n\to\infty}R \;=\;\lim_{n\to\infty}\,\frac{1}{8}\cdot\frac{\fra c{1}{5^n} + 5}{\frac{1}{5^n} + 1} \;=\;\frac{1}{8}\cdot\frac{0+5}{0+1} \;=\;\frac{5}{8}$

I actually did exactly that just before you posted and 5/8 isn't the answer either. I tried .625 too and it's not taking it. I can't figure out what is wrong.

5. Originally Posted by mollymcf2009
I actually did exactly that just before you posted and 5/8 isn't the answer either. I tried .625 too and it's not taking it. I can't figure out what is wrong.
The result of ratio test you got tells you its convergent. It doesn't tell you what the value of the sum is.

Do what o_O suggested and break it up into two geometric series.

6. Originally Posted by Chris L T521
The result of ratio test you got tells you its convergent. It doesn't tell you what the value of the sum is.

Do what o_O suggested and break it up into two geometric series.
Good lord, my brain has officially turned to mush...