Consider the generating function
$$f(x) = \sum_{k=0}^{\infty} F_k x^k$$
where $F_k$ are the Fibonacci numbers. The Fibonacci recurrence $F_k = F_{k-1} + F_{k-2}$ gives
$$f(x) = x + \sum_{k=2}^{\infty} (F_{k-1} + F_{k-2}) x^k = x + \sum_{k=1}^{\infty} F_k x^{k+1} + \sum_{k=0}^{\infty} F_k x^{k+2} = x + (x + x^2) f(x).$$
It follows that $(1 - x - x^2) f(x) = x$, so
$$f(x) = \frac{x}{1 - x - x^2}.$$
Substituting $x = \frac{1}{10}$, we conclude that
$$\sum_{k=0}^{\infty} \frac{F_k}{10^k} = \frac{10}{89}.$$
Similarly, substituting $x = - \frac{1}{10}$, we conclude that
$$\sum_{k=0}^{\infty} (-1)^k \frac{F_k}{10^k} = - \frac{10}{109}.$$
Generating functions are a very powerful method for understanding many sequences in combinatorics and other areas of mathematics. In this example we can use the generating function to go even further: via partial fraction decomposition we can quickly deduce Binet's formula
$$F_k = \frac{\phi^k - \varphi^k}{\phi - \varphi}$$
for the Fibonacci numbers, where $\phi, \varphi$ are the two roots of $x^2 = x + 1$, and this idea generalizes to other sequences defined by a linear recurrence.
A standard reference on generating functions is Wilf's generatingfunctionology.
