Unraveling a simple Inductive Equation [closed]

I have the following simple inductive equation:

$x_0 = n^k$

$x_1 = x_0 - {x_0}/{n^2}$

$x_i = x_{i-1} - {x_{i-1}}/{n^2}$

The question is - how can I represent $x_i$ in a non inductive manner? Specifically, I'm interested in figuring out whether for some constant $c$ independent of $n$ and $k$, $x_{n^c} < n^c$.

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This is not considered appropriate for this forum. Check the FAQ. However, notice x_3 = (n^k)*(1 - 1/n^2)^3 and x_4 = (n^k)*(1 - 1/n^2)^4. I suspect the c you want does not exist. Gerhard "Ask Me About System Design" Paseman, 2011.02.24 – Gerhard Paseman Feb 24 2011 at 17:41
I liken it to finding an absolute constant c such that ln(n) < c/(2(k-c)). This is why I think there is no such c. Gerhard "Ask Me About System Design" Paseman, 2011.02.24 – Gerhard Paseman Feb 24 2011 at 17:45
You may want to consider asking on math.stackexchange.com – S. Carnahan Feb 24 2011 at 18:12