# Limit of recurrent sequence including factorials and tetration

Let the sequence $$(a_n)_{n \geq 1}$$ be defined by $$a_1 >0$$, and the recurrent relation $$a_n = 2a_{n-1}*\frac{(n-1)^{(n-1)}}{(n-1)!}$$

Then, what is $$\lim_{n\to\infty} \frac {{a_{n+1}}^{1/{n+1}}}{{a_n}^{1/n}}$$?

So far, I've managed to prove that $$\lim_{n\to\infty}{(\frac{a_{n+1}}{a_n})}^{1/n}=e,$$ by using the Stirling limit. Therefore, it should suffice to calculate $$\lim_{n\to\infty} {{a_{n+1}}^{\frac{-1}{n(n+1)}}}.$$

• We only have a power of the form $x^x$, I would omit the "tetration" in the title, although we have actually a simple case of it. – Peter Feb 23 '20 at 10:28

$$a_n = a_1 2^ n \prod_{1 \le k \le n} \frac{k^k}{k!}$$
• Hello, I've took notice of this fact already - however much of the mess did not simplify away... because of the different exponents of $a_{n+1}$ and $a_n$ in the limit. – Parallelism Alert Feb 23 '20 at 12:06
• What is $\lim_{n \to \infty}(n + 1)/n$? – vonbrand Feb 23 '20 at 14:05