0
$\begingroup$

Show that

$\lim\limits_{n\to+\infty}\frac{2^{n^3}}{n!5^{n^2}-n^n}=+\infty$

without using l'Hôpitals rule or any special means.

My first attempts were to use inequalities to approximate $n!$ and $n^n$ but I am really unsure what to do with $2^{n^3}$ and $5^{n^3}$. Any specific hints on how to transform this fraction?

1 Answers 1

2

Hint: Try to estimate the denominator from above. For example $n! \le n^n \le (2^{n})^n = 2^{(n^2)}$. The basic idea is that $2^{(n^3)}$ is "so large" that you have much place to be generous with your estimations.

  • 0
    Oh I did some stupid mistakes - I have been able to get the desired result!2012-11-05