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Math Help - Determining Value of a Series

  1. #1
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    Determining Value of a Series

    We have started covering series and sequences in my calculus 2 class and I am having a hard time getting the problems started. For example I am told to determine the exact valiue of the following series

    \sum_{i\ =\ 0}^\infty 5 (\frac{2}{3})^i

    How do I proceed to start and complete the problem?
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  2. #2
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    Quote Originally Posted by fishguts View Post
    We have started covering series and sequences in my calculus 2 class and I am having a hard time getting the problems started. For example I am told to determine the exact valiue of the following series

    \sum_{i\ =\ 0}^\infty 5 (\frac{2}{3})^i

    How do I proceed to start and complete the problem?
    You should have covered geometric series in your class. This is an infinite geometric series.

    In general, a series of the form \sum_{n=0}^\infty ar^n=\frac{a}{1-r} provided that -1<r<1

    This should be all you need to complete the problem
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  3. #3
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    Geometric Series

    Hello fishguts
    Quote Originally Posted by fishguts View Post
    We have started covering series and sequences in my calculus 2 class and I am having a hard time getting the problems started. For example I am told to determine the exact valiue of the following series

    \sum_{i\ =\ 0}^\infty 5 (\frac{2}{3})^i

    How do I proceed to start and complete the problem?
    I assume that you have covered Geometric Series, because this is one:

    \sum_{i\ =\ 0}^\infty 5 (\frac{2}{3})^i=5 + 5.\tfrac23 + 5.(\tfrac23)^2+5.(\tfrac23)^3+...

    which has first term a = 5, common ratio r = \tfrac23

    The sum to infinity is \frac{a}{1-r}.

    Can you complete it now?

    Grandad
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  4. #4
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    For geometric series with common ratio r<1,

     \sum^{\infty}_{0} ar^{i} = \frac {a}{1-r}
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  5. #5
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    so then after plugging in all the numbers I get 15. Is this correct? it seems too easy...
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  6. #6
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    Hello fishguts
    Quote Originally Posted by fishguts View Post
    so then after plugging in all the numbers I get 15. Is this correct? it seems too easy...
    Yep! That's all there is to it. 15 is the answer.

    Grandad
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