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Question Number 173417 by mathlove last updated on 11/Jul/22

Commented by kaivan.ahmadi last updated on 11/Jul/22

Q_4 .  =Σ_1 ^∞ a^(2n−1) −Σ_1 ^∞ a^(2n) +Σ_1 ^∞ ((x/3))^n   if −1<a<1 and −3<x<3 then it is  equal to  a(1−a^2 )−a^2 (1−a^4 )+(x/3)(1−(x/3))

$${Q}_{\mathrm{4}} . \\ $$$$=\underset{\mathrm{1}} {\overset{\infty} {\sum}}{a}^{\mathrm{2}{n}−\mathrm{1}} −\underset{\mathrm{1}} {\overset{\infty} {\sum}}{a}^{\mathrm{2}{n}} +\underset{\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{{x}}{\mathrm{3}}\right)^{{n}} \\ $$$${if}\:−\mathrm{1}<{a}<\mathrm{1}\:{and}\:−\mathrm{3}<{x}<\mathrm{3}\:{then}\:{it}\:{is} \\ $$$${equal}\:{to} \\ $$$${a}\left(\mathrm{1}−{a}^{\mathrm{2}} \right)−{a}^{\mathrm{2}} \left(\mathrm{1}−{a}^{\mathrm{4}} \right)+\frac{{x}}{\mathrm{3}}\left(\mathrm{1}−\frac{{x}}{\mathrm{3}}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

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