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
2 years, 7 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 $$S= \frac{3}{5}$$

- 2 years, 7 months ago

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

- 2 years, 1 month ago