n-0-1-n-F-n-7-n-F-n-denotes-Fibbonocci-sequence-

Question Number 122023 by Dwaipayan Shikari last updated on 13/Nov/20

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {F}_{{n}} }{\mathrm{7}^{{n}} }\:\:\:\:\:\:\left({F}_{{n}} \:{denotes}\:{Fibbonocci}\:{sequence}\right) \\ $$

Answered by mr W last updated on 13/Nov/20

$${F}_{{n}} =\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}\left(\phi^{{n}} −\psi^{{n}} \right)\:{with} \\ $$$$\phi=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\psi=\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$${S}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {F}_{{n}} }{\mathrm{7}^{{n}} } \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\left(−\frac{\phi}{\mathrm{7}}\right)^{{n}} −\left(−\frac{\psi}{\mathrm{7}}\right)^{{n}} \right] \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}\left(\frac{\mathrm{1}}{\mathrm{1}+\frac{\phi}{\mathrm{7}}}−\frac{\mathrm{1}}{\mathrm{1}+\frac{\psi}{\mathrm{7}}}\right) \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}\left(\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{14}}}−\frac{\mathrm{1}}{\mathrm{1}−\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{14}}}\right) \\ $$$$=\frac{\mathrm{14}}{\:\sqrt{\mathrm{5}}}\left(\frac{\mathrm{1}}{\mathrm{15}+\sqrt{\mathrm{5}}}−\frac{\mathrm{1}}{\mathrm{15}−\sqrt{\mathrm{5}}}\right) \\ $$$$=−\frac{\mathrm{7}}{\:\mathrm{55}} \\ $$

Commented by Dwaipayan Shikari last updated on 13/Nov/20

$${Typo}\:\:\frac{\mathrm{1}}{\mathrm{1}−\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{14}}}=\frac{\mathrm{14}}{\mathrm{15}−\sqrt{\mathrm{5}}} \\ $$

Commented by Dwaipayan Shikari last updated on 14/Nov/20

$${Thanking}\:{you} \\ $$

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