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I would like to approximate the positive root of the following equation $$ 2(p+1)x^p - px - 2 = 0 $$ where $p$ is an integer. We could use the formula $(1 - y)^p \approx 1 - py$ for $y$ small to get an approximation of root $x_0 \approx \frac{1}{2p+1}$. However, I believe that we can make a stronger approximation. Could you please suggest some ideas?

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    Have you tried Newton's method ? (http://en.wikipedia.org/wiki/Newton%27s_method)2011-06-16
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    That approximation doesn't seem to be accurate for large $p$. Did you mean $x_0 \approx 1 - \frac{1}{2p+1}$?2011-06-16
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    If you are going to use this as part of a program, it could be a very good idea to use different "formulas" for the initial approximation for various ranges of values of $p$, say small, medium, and large.2011-06-16

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