converge?
is a polynomial with integer coefficients, with 
odd. Suppose that 
for all 
Prove that 
is nonzero for all 
and 
define
if 
& 
if 
for a positive integer r, such that
where 
and 
are positive integers.
and 
Y by
Let 
be a positive root of the equation 
. Prove that the following sequence converges: 
be a positive root of the equation 
. Prove that the following sequence converges: 
This post has been edited 4 times. Last edited by aothatday, Apr 16, 2025, 2:39 PM
.
,
hence ![${{n}^{2}}(\sqrt[n]{a}-\sqrt[n+1]{a})={{n}^{2}}\sqrt[n+1]{a}\left( \frac{\sqrt[n]{a}}{\sqrt[n+1]{a}}-1 \right)={{n}^{2}}\sqrt[n+1]{a}\left( {{a}^{\frac{1}{n(n+1)}}}-1 \right)=$](http://latex.artofproblemsolving.com/d/f/e/dfe9afef2115ded44a29597baa9c9884a9b8f3e6.png)
![$=\frac{n}{n+1}\sqrt[n+1]{a}\frac{{{a}^{\frac{1}{n(n+1)}}}-1}{\frac{1}{n(n+1)}}\to \ln a=\ln ({{x}^{2}}+x+1)$](http://latex.artofproblemsolving.com/e/f/6/ef670bda07317faa7fd305ee1a089ef8d71ab387.png)
