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Math Help - Infinite series of geometric?

  1. #1
    Super Member Anonymous1's Avatar
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    Infinite series of geometric?

    Okay, so I'm trying to find the constant c, but I am rusty on my elementary calculus. Any help is greatly appreciated.

    Necessarily,

    [c + c + (\frac{2}{3})^0 c+ (\frac{2}{3})^1c + (\frac{2}{3})^2 c.....] = 1

    \Rightarrow [2c + \sum_{n=2}^{\infty} (\frac{2}{3})^{n-2}c] = 1

    \Rightarrow [2 + \sum_{n=2}^{\infty} (\frac{2}{3})^{n-2}] = \frac{1}{c}

    Now, I need to find the sum... Can I reparameterize and let n-2 = j and then find the geo-sum? Like,

    \Rightarrow [2 + \sum_{j=0}^{\infty} (\frac{2}{3})^{j}] = 2+ \frac{1}{1- 2/3} = \frac{1}{c}

    \Rightarrow c= \frac{1}{5}?

    Thanks.
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  2. #2
    Junior Member
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    Quote Originally Posted by Anonymous1 View Post
    Okay, so I'm trying to find the constant c, but I am rusty on my elementary calculus. Any help is greatly appreciated.

    Necessarily,

    [c + c + (\frac{2}{3})^0 c+ (\frac{2}{3})^1c + (\frac{2}{3})^2 c.....] = 1

    \Rightarrow [2c + \sum_{n=2}^{\infty} (\frac{2}{3})^{n-2}c] = 1

    \Rightarrow [2 + \sum_{n=2}^{\infty} (\frac{2}{3})^{n-2}] = \frac{1}{c}

    Now, I need to find the sum... Can I reparameterize and let n-2 = j and then find the geo-sum? Like,

    \Rightarrow [2 + \sum_{j=0}^{\infty} (\frac{2}{3})^{j}] = 2+ \frac{1}{1- 2/3} = \frac{1}{c}

    \Rightarrow c= \frac{1}{5}?

    Thanks.

    yep your answer is correct, you can definitely sub in j=n-2 and as n-2 goes to infinity so will j.
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