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Math Help - Geometric Sum

  1. #1
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    Geometric Sum

    Hello all,

    Just a quick question, if you are summing r^t (where r is between zero and one, and you are summing across t to infinity), this converges to 1/(1-r). BUT if you don't start at 0, but you start at some value of t (lets call it t*), is there any nice way to express what this converges to?

    Thanks for the help,

    Nick
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  2. #2
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    Quote Originally Posted by salohcin View Post
    Hello all,

    Just a quick question, if you are summing r^t (where r is between zero and one, and you are summing across t to infinity), this converges to 1/(1-r). BUT if you don't start at 0, but you start at some value of t (lets call it t*), is there any nice way to express what this converges to?

    Thanks for the help,

    Nick
    |r| < 1

    \sum_{t = 0}^{\infty} r^t = 1 + r + r^2 + ... = \frac{1}{1-r}

    \sum_{t = 1}^{\infty} r^t = r + r^2 + ... = \frac{1}{1-r} - 1 = \frac{r}{1-r}
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  3. #3
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    Quote Originally Posted by salohcin View Post
    if you are summing r^t (where r is between zero and one, and you are summing across t to infinity), this converges to 1/(1-r). BUT if you don't start at 0, but you start at some value of t (lets call it t*), is there any nice way to express what this converges to?
    If N\in\mathbb{Z}~\&~|r|< 1 then \sum\limits_{k = N}^\infty  {r^k }  = \frac{{~r^N }}{{1 - r}}
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