Evaluate $\lim_{n \to \infty}\left(\frac{1^{1/x}+2^{1/x}+\ldots+n^{1/x}}{n}\right)^{nx}$

Question

$$\lim_{n \to \infty}\left(\frac{1^{1/x}+2^{1/x}+\ldots+n^{1/x}}{n}\right)^{nx}$$

I don't know any format or can't think of anything to solve this limit. It looks like it is Riemann's Sum Form but there is an x, so I am confused. Please help out. Thank You!

@ParasKhosla very sure. It is an IIT Question of previous year — rash, May 23 '20 at 07:42
Solution highly depends on $x$, say for $x=1$, limit is Infinite — Sahil Kumar, May 23 '20 at 07:47
I think this limit is undefined or something. I have tried logging this expression to simplify. Ultimately, I get a Riemann's sum but multiplied with another limit which is certainly undefined. — rash, May 23 '20 at 07:47
Has anyone verified this analytically? It seems to me that there is some typo but it would be great if anyone could confirm it — user600016, May 23 '20 at 10:10
Putting $x=1$ the expression becomes $((n+1)/2)^n$ which tends to $\infty $. Please fix typo in your question. Also remember that typos are more common in printed matter than most students think. — Paramanand Singh, May 23 '20 at 12:54

score 3 · Accepted Answer · answered May 23 '20 at 15:13

Lemma 1:

If the limit of $(a_n)_{n\in\mathbb N}$ exists or is infinite, then $$\lim_{n\to\infty}a_n=\lim_{n\to\infty}\frac1n\sum_{k=1}^na_k$$. Intuitively, the limit on the right is the average value of $(a_n)_{n\in\mathbb N}$, which is its limit. See here for more details.

Lemma 2:

If the limit of $(a_n)_{n\in\mathbb N}$ is $\infty$, then $$\lim_{n\to\infty}(a_n)^n=\infty$$.

Now it remains to see from lemma 1 that

$$\infty=\lim_{n\to\infty}n=\lim_{n\to\infty}\left(\frac1n\sum_{k=1}^nn^{1/x}\right)^x$$

and so the limit is $\infty$ for any $x\ne0$ by lemma 2.

Evaluate $\lim_{n \to \infty}\left(\frac{1^{1/x}+2^{1/x}+\ldots+n^{1/x}}{n}\right)^{nx}$

1 Answers1