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

$$\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n{F}_n}{7^n}\left(F_{n}denotesFibbonoccisequence\right)$$

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

$$F_n=\frac{1}{\sqrt{5}}({\varphi }^n-{\psi }^n)with$$$$\varphi =\frac{1+\sqrt{5}}{2}$$$$\psi =\frac{1-\sqrt{5}}{2}$$$$S=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n{F}_n}{7^n}$$$$=\frac{1}{\sqrt{5}}\underset{n=0}{\sum ^{\mathrm{\infty }}}[{(-\frac{\varphi }{7})}^n-{(-\frac{\psi }{7})}^n]$$$$=\frac{1}{\sqrt{5}}(\frac{1}{1+\frac{\varphi }{7}}-\frac{1}{1+\frac{\psi }{7}})$$$$=\frac{1}{\sqrt{5}}(\frac{1}{1+\frac{1+\sqrt{5}}{14}}-\frac{1}{1-\frac{\sqrt{5}-1}{14}})$$$$=\frac{14}{\sqrt{5}}(\frac{1}{15+\sqrt{5}}-\frac{1}{15-\sqrt{5}})$$$$=-\frac{7}{55}$$

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

$$Typo\frac{1}{1-\frac{\sqrt{5}-1}{14}}=\frac{14}{15-\sqrt{5}}$$

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

$$Thankingyou$$

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