Just a little question. When i have even power, it is obvious, for example: $$\sum _{n=0}^{\infty }\:x^{2n}\:=\:\frac{1}{1-x^2}$$ Is it correct to say that (when the power is odd): $$\sum _{n=0}^{\infty }\:x^{2n+1}\:=\sum_{n=0}^{\infty \:}\:x^{2n}\cdot x\:=\:\:x\cdot \sum_{n=0}^{\infty \:\:}\:x^{2n}\:=\:\frac{x}{1-x^2} $$
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See https://en.wikipedia.org/wiki/Geometric_progression – lab bhattacharjee Jul 07 '15 at 08:47
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it seems to be correct. you can check it by geometric sequences sit seems to be correct. you can check it by geometric sequences summation:$$a_1+a_1q+a_1q^2+...=\frac{a_1}{1-q}\|q|<1$$ then put $a_1=x,q=x^2$ – Khosrotash Jul 07 '15 at 08:49
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2It is exactly correct.(we can use properties of summation). – amir bahadory Jul 07 '15 at 08:51