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Math Help - [SOLVED] Infinite series #3

  1. #1
    Senior Member mollymcf2009's Avatar
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    [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!
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  2. #2
    o_O
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    \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?
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    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}

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  4. #4
    Senior Member mollymcf2009's Avatar
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    Quote Originally Posted by Soroban View Post
    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.
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  5. #5
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by mollymcf2009 View Post
    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.
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  6. #6
    Senior Member mollymcf2009's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    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...

    Answer is: 76/42

    Thanks y'all!!
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