A Fibonacci power sum

HardBrainteasers~7m

Evaluate $\displaystyle\sum_{k=0}^{\infty} \frac{F_k}{10^{k+1}}$ , where $F_0 = 0$ , $F_1 = 1$ , and $F_k = F_{k-1} + F_{k-2}$ . Express the answer as $\tfrac1M$ for an integer $M$ and find $M$ .

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Fibonacci generating function: $\sum_{k\ge0}F_k x^k = x/(1-x-x^2)$ .
Factor out $1/10$ and evaluate at $x=1/10$ : the sum is $\tfrac1{10}\cdot\tfrac{10}{89}$ .

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A Fibonacci power sum

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