Fibonacci numbers as decimal digits

Let $(F_k)$ be the Fibonacci sequence starting from zero and one, where every later term is the sum of the two before it ($0,1,1,2,3,5,\dots$). Compute $S=\sum_{k=0}^{\infty}\frac{F_k}{10^{k+1}}.$

$1/89$. Use the generating function $G(x)=\sum F_kx^k=\dfrac{x}{1-x-x^2}$. Then $S=\tfrac1{10}G\!\left(\tfrac1{10}\right)=\tfrac1{10}\cdot\dfrac{0.1}{1-0.1-0.01}=\dfrac{0.01}{0.89}=\tfrac1{89}$.