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Let be a polynomial with real coefficients. Calculate the value of this limit:

$$\lim_{n \rightarrow \infty} |P(1)...P(n+1)|^ \frac1{n+1}-|P(1)...P(n)|^ \frac1{n} $$

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    Hint: first try it with $P(x)=ax^m$, the leading term. You can later argue that the other terms don't matter as $n \to \infty$2012-05-25
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    It seems the obvious first thing to try is to replace $P$ with its behavior "at $+\infty$": i.e. $P(x) \sim a x^k$ for some integer $k$ and real number $a$. Then see if you can compute this limit, and prove it equal to the original. Disclaimer: I haven't carried out the calculation yet.2012-05-25
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    If $P$ have zero at integer points then the limit is zero. On the other hand, for $P(n)=n$ the limit is $1/e$.2012-05-25
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    @Ross Millikan: having received these information it means that the limit seems to be $\frac{|a|}{e^{m}}$ for m=1, and infinity for m>1. I wonder how i can argue that the other terms don't matter.2012-05-25
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    Note that for $P(n)=(n-1)(n-2)(n-3)$ the limit is zero2012-05-25
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    @Chris The apparent difficulty of this problems is the number of absract entities. Say the polynomial is $$P(x)=\sum_{j=0}^k a_j x^j$$ and $P$ has no integer roots. Then the product inside the root is $$\mathcal P(n) =\prod_{m=1}^n \sum_{j=0}^k a_j m^j$$ The limit you want is $$|\mathcal P(n+1)|^{\frac 1 {n+1}}-|\mathcal P(n)|^{\frac 1 n}$$ Try to find the limit for simple polynomials (deg. 1,2, maybe 3) and then try to conjecture something.2012-06-04

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