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Thread: sum of power series

  1. #1
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    sum of power series

    This might seem easy, but could anyone share the trick how to do the following sums?
    the sum of 8^n/(3n)!, n goes from 0 to infinity
    the sum of 27^n/(3n+1)!, n goes from 0 to infinity
    Any input is appreciated!
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  2. #2
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    Quote Originally Posted by nngktr View Post
    This might seem easy, but could anyone share the trick how to do the following sums?
    the sum of 8^n/(3n)!, n goes from 0 to infinity
    Let \omega be a complex cube root of 1. Then \sum_{n=0}^\infty\frac{2^n}{n!} = e^2,\quad \sum_{n=0}^\infty\frac{(2\omega)^n}{n!} = e^{2\omega},\quad \sum_{n=0}^\infty\frac{(2\overline{\omega})^n}{n!} = e^{2\overline{\omega}}.

    However, 1^n+\omega^n + \overline{\omega}^{\,n} is equal to 3 if n is a multiple of 3, and 0 otherwise.

    Therefore \sum_{n=0}^\infty\frac{8^n}{(3n)!} = <br />
\tfrac13\bigl(e^2 + e^{2\omega} + e^{2\overline{\omega}}\bigr). The expression on the right should obviously be real, so you need to plug in the value for \omega and check that this is indeed the case.
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