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Math Help - How to estimate summations

  1. #1
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    How to estimate summations

    i.e.

    \sum_{n = 0}^{\infty}\frac{1}{2^{n}} = \frac{1}{2^{0}} +  \frac{1}{2^{1}} + \frac{1}{2^{2}} + \frac{1}{2^{3}} + \frac{1}{2^{4}} +  \frac{1}{2^{5}} + \frac{1}{2^{6}} + \cdots = ~1.99138889...

    Is there a way you can know this solution is 2 without having to perform all of the calculations I did to find which number the sums are approaching? And is there a general method for questions like these to find the solution without having to perform a lot of calculations?
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  2. #2
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    Re: How to estimate summations

    Quote Originally Posted by daigo View Post
    i.e.

    \sum_{n = 0}^{\infty}\frac{1}{2^{n}} = \frac{1}{2^{0}} +  \frac{1}{2^{1}} + \frac{1}{2^{2}} + \frac{1}{2^{3}} + \frac{1}{2^{4}} +  \frac{1}{2^{5}} + \frac{1}{2^{6}} + \cdots = ~1.99138889...

    Is there a way you can know this solution is 2 without having to perform all of the calculations I did to find which number the sums are approaching? And is there a general method for questions like these to find the solution without having to perform a lot of calculations?
    Yes, this is an infinite geometric series with \displaystyle \begin{align*} a = 1, r = \frac{1}{2} \end{align*}, so the sum is \displaystyle \begin{align*} \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \end{align*}.
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  3. #3
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    Re: How to estimate summations

    The sum of a geometric series is given by

    \sum_{i=0}^{\infty} r^i = \frac{1}{1-r}, given |r| < 1. In this case, r = \frac{1}{2}.
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