I found f(x), now what?

Using the sum of an infinite geometric series, we know that $f(x)=\frac { x }{ x-1 }$ .

Note that $f(x)=\frac { x }{ x-1 }$ is the inverse of itself. Therefore, $f(\frac { x }{ x-1 } ) = f(f(x)) = f({ f }^{ -1 }(x)) = x$ .

Since $f(\frac { x }{ x-1 } )=10$ , we can say that $x=\boxed { 10 }$

you know that the first given summation is equal to x/x-1 by definition of geometric sum. You also know that f( x/x-1) is equal to 10. Note that you have now to important informations f( x/x-1)=10 and also f(x/x-1)=x( due to the properties of the inverse function). You can conclude that x=10 with a simple substitution. Hope i learn how to use latex soon.

Nice problem by the way.

g.p infinite sum = x/(x-1) =f(x) put x as x/(x-1) in f(x)=10