Problem 219 (difficulty: 4/10)

Let \(\displaystyle a_k\ne0\) and \(\displaystyle p(x)=a_0 +a_1 x+\ldots + a_k x^k\). Prove that

\(\displaystyle \lim\limits_{n\to +\infty}\frac{p(n+1)}{p(n)}=1. \)

Solution:

Simplify by \(\displaystyle a_0n^k\):

\(\displaystyle \frac{p(n+1)}{p(n)}=\frac{\left(1+\frac1n\right)^k+a(n)}{1+b(n)}, \)

where \(\displaystyle a(n)\to 0\) and \(\displaystyle b(n)\to 0\).

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