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Math Help - Rearranging Equation to determine conv/div

  1. #1
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    Rearranging Equation to determine conv/div

    I just found it simpler to take a screenshot. I have no idea how to go about rearranging the equation like the solution does below.

    Thanks.
    Attached Thumbnails Attached Thumbnails Rearranging Equation to determine conv/div-equation.jpg  
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  2. #2
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    Note that,
    \frac{\pi ^n}{3^{n+1}} = \frac{\pi^n}{3^n \cdot 3} = \frac{1}{3} \cdot \frac{\pi^n}{3^n} = \frac{1}{3} \cdot \left( \frac{\pi}{3} \right)^n
    Thus,
    \sum_{n=0}^{\infty}\frac{\pi^n}{3^{n+1}} = \sum_{n=0}^{\infty} \frac{1}{3}\cdot \left( \frac{\pi}{3} \right)^n = \frac{1}{3}\sum_{n=0}^{\infty} \left( \frac{\pi}{3} \right)^n
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  3. #3
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    Ahh, so there is a property that allows 3^(n+1)=3^n + 3^1

    Am I right on that?
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  4. #4
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    Quote Originally Posted by trev-gdr View Post
    Ahh, so there is a property that allows 3^(n+1)=3^n + 3^1

    Am I right on that?
    3^{n+1} = \underbrace{3\cdot 3\cdot ... \cdot 3}_{n+1 \text{ times }} = \underbrace{3\cdot 3\cdot ... \cdot 3}_{n\text{ times }}\cdot 3 = 3^n \cdot 3.

    In general, x^{n+m} = x^n \cdot x^m where n,m are positive integers.

    You supposed to have known this in algebra. Not when learning Calculus.
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  5. #5
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    Sorry I put + and thought *.

    And it does so happen that people tend to forget things when they haven't done it in a while....hmm what an odd theory .

    Thanks!
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