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How do I find the explicit formula for the following summation: $x^1 + x^3 + x^5 + ... $

I know $1 + x + x^2 + x^3 + ... = 1/(1-x)$, but this is quite a different series.

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$$x+x^3+x^5+....=x(1+x^2+x^4+....)=x(1+(x^2)^1+(x^2)^2+(x^2)^3+....)=\frac {x}{1-x^2}$$

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  • $\begingroup$ Thanks! why didnt I think of this?! $\endgroup$ – AbcXYZ Dec 7 '12 at 11:19
  • $\begingroup$ So clever, Amr! :-) $\endgroup$ – 000 Dec 7 '12 at 11:29
  • $\begingroup$ Thank you. But I am not that clever really for posting this answer $\endgroup$ – Amr Dec 7 '12 at 11:30
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Not much different,$$x^1+x^3+x^5\cdots=x(1+x^2+(x^2)^2+\cdots)$$

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  • $\begingroup$ Edited the formula, as it was not correct... $\endgroup$ – Jan Keersmaekers Dec 7 '12 at 11:06

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