Analytical solution for u in P(u) = u^n(1-u)/(1-u^{n+1}) = x

[mkkrnfoo85]

So, here's the motivation to my problem:

I have this following equation:

$$P(u) = \frac{u^n}{1+u+u^2+...+u^n}$$

, 'n' being a given constant.

I figured out that I could simplify the power series on the denominator, and get a better solution for P(u):

$$P(u) = \frac{u^n-u^{n+1}}{1 - u^{n+1}}$$Now, I am interested in P(u) = x, and 'x' is a number between 0 and 1.
What I have been struggling with is finding an analytical solution for u(x).
To make it more clear, I would like to know what 'u' will equal, for a given 'x' value, and a known 'n'.

I've tried a lot of different approaches, but I just can't seem to come up with an analytical solution for u(x).

One attempt and where I get stuck: $$u^n + u^{n+1}(x-1) = x$$Any help would be appreciated. I would like to be enlightened. =p

Thanks,

Mark

 

[Gib Z]

I am inclined to say there is no analytical solution for it, though I have no proof, just instinct.

 

[slider142]

You can use the inverse function theorem along with http://planetmath.org/encyclopedia/DerivativeOfInverseFunction.html to find the derivative of u(x) from what you have (x(u)).