2
$\begingroup$

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} $

  • 0
    @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

0 Answers 0