As $n$ increases, the value of $\sqrt[ne^{\pi}]{\pi}$ approaches $1$. Is there a name for this result?

Question

I was messing around with $\pi$ and $e$ on the Desmos calculator, and came up with the observation that this value:

$$\sqrt[ne^{\pi}]{\pi}$$

approaches $1$ as $n$ increases.

(To be clear, that's the $(ne^{\pi})$-th root of $\pi$.)

After around a half hour of looking on the internet and Wikipedia, I have yet to find anything explaining this. Is this something relevant to math (is it important)? If so, what is its name? I am making a compilation of cool things one can do with $\pi$, and would love to include a source/name for it.

Thank you, and have a good day!

Answer 1

Mathematically speaking, this phenomenon could be expressed by writing

$$\lim_{n \to \infty} \pi^{1/(ne^\pi)} = 1$$

As it turns out, this phenomenon is not unusual -- for all positive real numbers $x$,

$$\lim_{n \to \infty} x^{1/n} = 1$$

It should be fairly evident how this might apply to your case, since, for example, $\pi^{1/(ne^\pi)} = \left( \pi^{1/e^\pi} \right)^{1/n}$. The curiosity is, how to prove this? Well, since the exponential in this case is continuous, we can claim

$$\lim_{n \to \infty} x^{1/n} = x^{\lim_{n \to \infty} 1/n}$$

Obviously, as $n$ grows without bound, $1/n$ approaches $0$, so the exponent approaches zero, and thus your limit approaches $x^0 = 1$.

Answer 2

$$a_n=\sqrt[ne^{\pi}]{\pi}\implies\log(a_n)=\frac 1{ne^{\pi}}\log(\pi)=\frac{\log(\pi) }{e^\pi }\frac 1n$$ So, $\log(a_n)\to0$ and then $a_n\to 1$