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Math Help - [SOLVED] sum of series

  1. #1
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    [SOLVED] sum of series

    I am supposed to tell whether this converges or diverges and if it converges tell what it converges to. I'm unclear on how to do the last part.

    the summation from n=0 to infinity of 3^(n+1)/4^(n-2)

    I used the ratio test and found that it converges. I don't know what to do now.
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  2. #2
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    It is equal to 3\cdot4^2 + 3^2 \cdot 4 + \sum_{n=2}^\infty 3^3(\frac34)^{n-2}.
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  3. #3
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    Quote Originally Posted by maddas View Post
    It is equal to 3\cdot4^2 + 3^2 \cdot 4 + \sum_{n=2}^\infty 3^3(\frac34)^{n-2}.
    what?
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  4. #4
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    I don't know anything about this. And no matter how many different ways I ask this I just get answers that are too complicated for me to understand. I have to figure this out by 9 am. And I don't have a clue as to what I'm doing.
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  5. #5
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    Well, write it all out.

    3\cdot 4^2 + 3^2\cdot 4 + 3^3\cdot 1 + {3^4\over 4} + {3^5\over 4^2} + {3^6\over 4^3} +\cdots

    Now factor the later part like

    3\cdot 4^2 + 3^2\cdot 4 + 3^3\Big(1 + {3\over 4} + {3^2\over 4^2} + {3^3\over 4^3} +\cdots\Big)

    The sum in parenthesis is geometric. You should know that 1+r+r^2+...=1/(1-r) for |r|<1. So in this case

    3\cdot 4^2 + 3^2\cdot 4 + 3^3\cdot{1\over 1-\frac34}

    Now what does this expression equal?

    Get it? You probably shouldn't wait until the morning before to figure this stuff out :/
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  6. #6
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    Quote Originally Posted by maddas View Post
    Well, write it all out.

    3\cdot 4^2 + 3^2\cdot 4 + 3^3\cdot 1 + {3^4\over 4} + {3^5\over 4^2} + {3^6\over 4^3} +\cdots

    Now factor the later part like

    3\cdot 4^2 + 3^2\cdot 4 + 3^3\Big(1 + {3\over 4} + {3^2\over 4^2} + {3^3\over 4^3} +\cdots\Big)

    The sum in parenthesis is geometric. You should know that 1+r+r^2+...=1/(1-r) for |r|<1. So in this case

    3\cdot 4^2 + 3^2\cdot 4 + 3^3\cdot{1\over 1-\frac34}

    Now what does this expression equal?

    Get it? You probably shouldn't wait until the morning before to figure this stuff out :/
    I just got the assignment Fri and I've been trying to figure this out all weekend. And it sorta makes sense. I've never learned about geometric sequences.
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