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Infinite sequence

Given the sequence \(\frac{1}{3}+\frac{1}{9}+\frac{2}{27}+\frac{3}{81}+.....\frac{F_k}{3^k}+......\) where \(F_k\) is the Fibonacci sequence. Compute the sum.

Note by William Isoroku
1 year, 10 months ago

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Here, Let S be its sum.

So, \( {S} - \frac{S}{3} = \frac{1}{3} - \frac{1}{3^2} {(S)} \)

Or \( S= \frac{3}{5} \) Sachin Vishwakarma · 1 year, 10 months ago

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Use this \[\sum_{k=1}^{\infty} F_k x^k = \frac{x}{1-x-x^2} \quad \forall |x| < \frac{\sqrt5-1}{2}\]

@William Isoroku Deeparaj Bhat · 1 year, 4 months ago

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