# Help: want to find an analytical soln to this =p

1. Nov 7, 2007

### 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

Last edited: Nov 7, 2007
2. Nov 7, 2007

### Gib Z

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

3. Nov 7, 2007

### slider142

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

Last edited by a moderator: May 3, 2017