7. [5] Let $F_{n}$ be the Fibonacci sequence, that is, $F_{0}=0, F_{1}=1$, and $F_{n+2}=F_{n+1}+F_{n}$. Compute $\sum_{n=0}^{\infty} F_{n} / 10^{n}$.
Answer:
$10 / 89$
Solution: Write $F(x)=\sum_{n=0}^{\infty} F_{n} x^{n}$. Then the Fibonacci recursion tells us that $F(x)-x F(x)-$ $x^{2} F(x)=x$, so $F(x)=x /\left(1-x-x^{2}\right)$. Plugging in $x=1 / 10$ gives the answer.