# $\lim_{n\to\infty} \underbrace{\int_{0}^{1}\cdots \int_{0}^{1}}_{n}\frac{x_1^{505}+\cdots +x_n^{505}}{x_1^{2020}+\cdots +x_n^{2020}}dx_1\cdots dx_n$

Evaluate this multiple integral inside a limit:

$$\lim_{n\to\infty} \underbrace{\int_{0}^{1}\cdots \int_{0}^{1}}_{n}\frac{ \sum _{k=1}^{n}x_k^{505}}{\sum_{k=1}^{n}x_k^{2020}} \mathrm d x_1\cdots \mathrm dx_n$$

Someone sent me this question and apparently it was from a very hard competition/exam. I couldn't evaluate this with any techniques that I know of (my knowledge is very limited) and I have also posted this question on my Instagram account to see if anyone could evaluate it, but unfortunately still no one could evaluate it.

Since there are many people who are really good at mathematics here and it is relatively easier to gain attention here than Instagram, I would want to see if anyone can evaluate it. Probably this expression would need an advanced technique that I don't know of or an unexpected trick to evaluate. Hopefully I can learn something from this post!

Edit: some comments suggested that intuitively it would evaluate to $$\frac{2021}{506}$$ with similar techniques used in the evaluation of a similar integral: $$\lim_{n\to\infty} \underbrace{\int_{0}^{1}\cdots \int_{0}^{1}}_{n}\frac{ \sum _{k=1}^{n}x_k^{2}}{\sum_{k=1}^{n}x_k} \mathrm d x_1\cdots \mathrm dx_n=\frac{2}{3}$$ That question is linked to this one and it would be a good idea to have a look at the techinques used there to see if it could be implemented here.

More information: Now I think this integral doesn't involve any techniques in multivariable calculus but instead some techniques from probability theory (which I don't fully understand yet) but I really want to see the solution and learn something from it.

• This is not the same problem, but to anyone who tries to do it, a similar integral: mathoverflow.net/questions/288085/how-to-evaluate-this-integral ....maybe this may provide some help – Saket Gurjar May 19 at 12:03
• Following math.stackexchange.com/questions/128086/…, a good guess for the value is $\frac{2021}{506}$. However, interchanging the $\mathbb P$-limit and the expectation operator seems more difficult than in the linked question. – Václav Mordvinov May 19 at 12:40
• This question is stated in a rather confusing manner, because the actual question is about the integral itself. (The limit of the integral may be computed using the law of large numbers, although the rigorous proof might be a bit demanding.) OP should consider clarifying what is asked in this question. – Sangchul Lee May 19 at 12:45
• @ProfessorVector Good thing the limit doesn't care about small $n$, then. I believe the integral does converge when $n>1515$. – aschepler May 19 at 18:21
• Curious why this is off-topic or breaking other guidelines. I had a proof that the limit is in fact $\frac{2021}{506}$ ready to go. – aschepler May 19 at 18:33