Prove that $\frac{t^{x-1}}{1+t^y} = \sum_{n=0}^{\infty} (-1)^nt^{x-1+yn}$.

I tried expanding the exponential terms by substituting the definition of exp such that I have a series in the numerator and denominator but no matter how I tried manipulating the expressions, I wasn't able to show the equality.

Thanks in advance, whiterock.


$$\sum_{n=0}^{\infty} (-1)^nt^{x-1+yn} = \sum_{n=0}^{\infty} (-1)^nt^{x-1}(t^y)^n = t^{x-1}\sum_{n=0}^\infty(-t^y)^n = t^{x-1} \frac{1}{1-(-t^y)}=\frac{t^{x-1}}{1+t^y}$$ whenever the exponentials are well defined and $|t^y|<1$.

  • 1
    $\begingroup$ Thank you! Wouldn't have thought that in this direction it's so much easier than the other. One minor thing: you forget the "- 1" in the exponent right at the end :) $\endgroup$ – whiterock Nov 8 '18 at 20:46
  • $\begingroup$ Thank's and you're welcome. $\endgroup$ – Tito Eliatron Nov 8 '18 at 20:51

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