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Math Help - Closed Expression for Series

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    Closed Expression for Series

    By E I mean sum from k=1 to n

    Find a closed expression for

    1+ E [1 / 2^(k-1)]

    I used the formula for geo series and have the sum = 1 - 1/2^n. Is this correct? How do I make a closed expression?
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    Re: Closed Expression for Series

    Quote Originally Posted by veronicak5678 View Post
    By E I mean sum from k=1 to n

    Find a closed expression for

    1+ E [1 / 2^(k-1)]

    I used the formula for geo series and have the sum = 1 - 1/2^n. Is this correct? How do I make a closed expression?
    You have \displaystyle 1 + \sum_{k = 1}^n{\frac{1}{2^{k-1}}} = 1 + \sum_{k = 1}^n{\left(\frac{1}{2}\right)^{k - 1}}

    This sum is an infinite geometric series with \displaystyle a = 1, r = \frac{1}{2}. Since \displaystyle |r| < 1, the sum is equal to \displaystyle \frac{a}{1 - r}. Go from here.
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    Re: Closed Expression for Series

    I thought that formula was for infinite series only? Can I use it if I go from 1 to n?
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    Re: Closed Expression for Series

    Quote Originally Posted by veronicak5678 View Post
    I thought that formula was for infinite series only? Can I use it if I go from 1 to n?
    \sum\limits_{k = 1}^n {r^k }  = \frac{{r - r^{n + 1} }}{{1 - r}}
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    Re: Closed Expression for Series

    Quote Originally Posted by veronicak5678 View Post
    I thought that formula was for infinite series only? Can I use it if I go from 1 to n?
    Yes you are right. But there is a formula for finite geometric series as well, as Plato has explained.
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    Re: Closed Expression for Series

    Using that formula for finite series is how I came up with the expression from my first post, 1 - 1/2^n. Is this correct? Is it closed?
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    Re: Closed Expression for Series

    Quote Originally Posted by veronicak5678 View Post
    Using that formula for finite series is how I came up with the expression from my first post, 1 - 1/2^n. Is this correct? Is it closed?
    Yes it's correct. It's now closed because it is not a sum depending on k, it is a single value.
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    Re: Closed Expression for Series

    OK. Thanks to all for helping.
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