# Evaluate $\lim\limits_{n\to\infty}\left(\sqrt [n+1]{\frac {a_{n+1}}{b_{n+1}}}-\sqrt[n]{\frac {a_n}{b_n}}\ \right)$

Consider two sequences $$(a_n)_{n\ge 1}$$ and $$(b_n)_{n\ge 1}$$ of positive real numbers such that

$$\lim_{n\to\infty} \frac {a_{n+1}}{n^2\cdot a_n}=x>0$$

and

$$\lim_{n\to\infty} \frac {b_{n+1}}{n\cdot b_n}=y>0$$

Evaluate:

$$\lim_{n\to\infty}\left(\sqrt [n+1]{\frac {a_{n+1}}{b_{n+1}}}-\sqrt[n]{\frac {a_n}{b_n}}\ \right)$$

My thoughts: Intuitively $$\dfrac{a_{n+1}}{a_n} \sim xn^2$$ and $$\dfrac{b_{n+1}}{b_n} \sim yn$$, so $$a_n \sim x \sqrt[n]{(n!)^2}$$ and $$b_n \sim y\sqrt[n]{n!}$$. Since $$\lim\limits_{n\to \infty} \dfrac{n}{\sqrt[n]{n!}} = e$$, we should get:

$$\sqrt[n+1]{\frac{a_{n+1}}{b_{n+1}}} \sim \frac{x}{y}\cdot \sqrt[n+1]{(n+1)!} \sim\frac{x}{y}\cdot \frac{n+1}{e}$$

and the final answer should be $$\dfrac{x}{ye}$$. But how do we write this with sound arguments?

I think I completed a proof thanks to Gary's idea with the $$c_n$$ substitution and the ratio limits. Bear with me. First, let's make a little different substitution $$c_n = \sqrt[n]{\dfrac{a_n}{b_n}}$$ and use the following criterion.

Lemma (Cauchy-d'Alembert): If $$x_n > 0,\ (\forall) n \in \mathbb{N}$$ and the limit $$l = \lim\limits_{n\to \infty} \dfrac{x_{n+1}}{x_n}$$ exists, then $$\sqrt[n]{x_n}$$ converges to $$l$$.

Since:

$$\lim_{n \to \infty}\frac{\dfrac{a_{n+1}}{(n+1)^{n+1}\cdot b_{n+1}}}{\dfrac{a_n}{n^n\cdot b_n}} = \lim_{n \to \infty} \left(\frac{a_{n+1}}{n^2\cdot a_n} \cdot \frac{n\cdot b_n}{b_{n+1}} \cdot \frac{n^{n+1}}{(n+1)^{n+1}}\right) = \frac{x}{y \cdot e}$$

we can apply Cauchy D'Alembert for

$$\dfrac{c_n}{n} = \sqrt[n]{\frac{a_n}{n^n\cdot b_n}} \to \frac{x}{y \cdot e}$$

From this we also have

$$\lim\limits_{n\to\infty}\, \frac {c_{n+1}}{c_n}=\lim\limits_{n\to\infty}\, \left[\frac {c_{n+1}}{n+1}\cdot\frac n{c_n}\cdot\frac {n+1}n\right]=1$$

and

$$\lim\limits_{n\to\infty}\, \left(\frac {c_{n+1}}{c_n}\right)^n=\lim\limits_{n\to\infty}\, \left(\frac {a_{n+1}}{n^2\cdot a_n}\cdot\frac {n\cdot b_n}{b_{n+1}}\cdot n \cdot \sqrt[n]{\frac {b_n}{a_n}}\right)^{\frac n{n+1}}=\frac xy\cdot\frac {y\cdot e}x=e$$

Therefore, we can compute the required limit as follows:

\begin{aligned} \lim_{n\to\infty}\, \left(\sqrt[n+1]{\frac {a_{n+1}}{b_{n+1}}}-\sqrt[n]{\frac {a_n}{b_n}}\ \right) &= \lim_{n\to\infty}\, \left(c_{n+1}-c_n\right) \\ &=\lim_{n\to\infty}\, \left[c_n\cdot\left(\frac {c_{n+1}}{c_n}-1\right)\right]\\ &= \lim_{n\to\infty}\ \left[\frac {c_n}n\cdot\frac {e^{\ln\left(\dfrac {c_{n+1}}{c_n}\right)}-1}{\ln\left(\dfrac {c_{n+1}}{c_n}\right)}\cdot\ln\left(\frac {c_{n+1}}{c_n}\right)^n\right] \\ &= \frac x{y\cdot e}\cdot 1\cdot\ln e \\ &=\frac x{y\cdot e} \end{aligned}

This completes the solution. And interesting remark is that choosing $$a_n= (n!)^2$$ and $$b_n = n!$$, we get the following limit (Lalescu):

$$\lim_{n\to\infty}\, \left(\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}\right)=\frac 1{e}$$

Some observations. Define $$c_n : = \frac{1}{n!}\frac{a_n}{b_n}.$$ Then by the assumptions on $$a_n$$ and $$b_n$$, we have $$\mathop {\lim }\limits_{n \to + \infty } \frac{c_{n + 1}}{c_n} = \frac{x}{y}\quad \text{ and hence }\quad \mathop {\lim }\limits_{n \to + \infty } \sqrt[n]{c_n} = \frac{x}{y}.$$ Note that $$\sqrt[n]{\frac{a_n}{b_n}} = \sqrt[n]{n!c_n}.$$ By Stirling's formula $$\sqrt[n]{{n!}} = \frac{n}{e} + \frac{{\log (2\pi n)}}{{2e}} + \mathcal{O}\left( {\frac{1}{n}} \right).$$ If it was true that $$\sqrt[n]{{c_n }} = \frac{x}{y} +\frac{K}{n}+ o\left( {\frac{1}{n}} \right)$$ then we would have $$\sqrt[{n + 1}]{{(n + 1)!c_{n + 1} }} - \sqrt[n]{{n!c_n }} = \frac{x}{y}\frac{1}{e} + \mathcal{O}\left( {\frac{1}{n}} \right),$$ meaning that the limit is indeed $$\frac{x}{y}\frac{1}{e}$$. In any case, $$\mathop {\lim \inf }\limits_{n \to + \infty } \left( {\sqrt[{n + 1}]{{\frac{{a_{n + 1} }}{{b_{n + 1} }}}} - \sqrt[n]{{\frac{{a_n }}{{b_n }}}}} \right) = \mathop {\lim \inf }\limits_{n \to + \infty } \frac{{\sqrt[{n + 1}]{{\frac{{a_{n + 1} }}{{b_{n + 1} }}}} - \sqrt[n]{{\frac{{a_n }}{{b_n }}}}}}{{n + 1 - n}} \le \mathop {\lim }\limits_{n \to + \infty } \frac{{\sqrt[n]{{n!c_n }}}}{n} = \frac{x}{{ye}} \\ \le \mathop {\lim \sup }\limits_{n \to + \infty } \frac{{\sqrt[{n + 1}]{{\frac{{a_{n + 1} }}{{b_{n + 1} }}}} - \sqrt[n]{{\frac{{a_n }}{{b_n }}}}}}{{n + 1 - n}} = \mathop {\lim \sup }\limits_{n \to + \infty } \left( {\sqrt[{n + 1}]{{\frac{{a_{n + 1} }}{{b_{n + 1} }}}} - \sqrt[n]{{\frac{{a_n }}{{b_n }}}}} \right).$$ Thus, if the limit in question exists, it must be $$\frac{x}{y}\frac{1}{e}$$.