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Math Help - Value of S?

  1. #1
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    Value of S?

    Could someone help with this, A hint is provided but I cannot seems to find a solution.
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    S = 1 + 0.5 + 0.25 + ... + (\frac{1}{2})^n

    What is the Value of S? [HINT = Multiply S by 2, then Solve]
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Air View Post
    Could someone help with this, A hint is provided but I cannot seems to find a solution.
    __________________________________________________ _______________________

    S = 1 + 0.5 + 0.25 + ... + (\frac{1}{2})^n

    What is the Value of S? [HINT = Multiply S by 2, then Solve]
    you have actually this
    S = 1 + \frac{1}{2} + \left( {\frac{1}{2}} \right)^2 + ... + \left( \frac{1}{2} \right)^n --------(a)

    if we multiply 2, we have

    2S = 2 + 1 + \frac{1}{2} + \left( {\frac{1}{2}} \right)^2 + ... + \left( \frac{1}{2} \right)^{n-1} ----------(b)

    subtracting (b) - (a), we have

    S = 2 - \left( \frac{1}{2} \right)^n, so if we take n to be a large number, then S \approx 2
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  3. #3
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    Quote Originally Posted by kalagota View Post
    S = 2 - \left( \frac{1}{2} \right)^n, so if we take n to be a large number, then S \approx 2
    What do you mean by n is larger number? Can you explain how the approximation of 2 is obtained?
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  4. #4
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    Quote Originally Posted by Air View Post
    What do you mean by n is larger number? Can you explain how the approximation of 2 is obtained?
    Hello,

    you are supposed to know that

    \lim_{n\to \infty}\left(\frac12\right)^n = 0

    With your problem you have:

    \lim_{n\to \infty}\left(2- \left(\frac12\right)^n \right)= 2-0 = 2
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