Calculate $\lim_{n\to\infty} \frac{ (1^{1^p}2^{2^p}\cdot...\cdot n^{n^p})^{ 1/n^{p+1} }}{n^{1/(p+1)}}$

Question

Calculate: $$\lim_{n\to\infty} \frac{ (1^{1^p}2^{2^p}\cdot...\cdot n^{n^p})^{ 1/n^{p+1} }}{n^{1/(p+1)}}$$

I've done some steps as follows: $$a_n:=\frac{ (1^{1^p}2^{2^p}\cdot...\cdot n^{n^p})^{ 1/n^{p+1} }}{n^{1/(p+1)}} \iff \ln a_n=\frac{1}{n^{p+1}}\big(\sum_{k=1}^nk^p\ln k-\frac{n^{p+1}}{p+1}\ln n\big) \iff \\\ln a_n =\frac{1}{n}\sum_{k=1}^n\big[\big(\frac{k}{n}\big)^p\ln \frac{k}{n}\big]+\frac{1}{n}\sum_{k=1}^n\big(\frac{k}{n}\big)^p\ln n-\frac{\ln n}{p+1}.$$ Then, I was wondering if I could make some integrals out of it but still there are some odd terms.

I think my approach isn't so good...

Answer 1

I seem to remember answering this question sometimes, but I didn't find it! So, I write the answer again, I didn't COPY my previous answer. Thank @metamorphy for pointing out this！The following is my previous answer. Computing limit of a product

$$\frac{1}{n}\sum_{k=1}^n\big[\big(\frac{k}{n}\big)^p\ln \frac{k}{n}\big] \to\int_{0}^{1}x^p\ln x dx.$$ is not difficult.

What you really need is the limit: $$\lim_{n\to\infty}\frac{1}{n}\left(\sum_{k=1}^n\big(\frac{k}{n}\big)^p\ln n-\frac{\ln n}{p+1}\right) =\lim_{n\to\infty}\left(\frac{1}{n}\sum_{k=1}^n\left(\frac{k}{n}\right)^p-\frac{1}{p+1}\right)\ln n=0.$$

To get this, we have the following result(https://math.stackexchange.com/a/149174/72031): Suppose $f'$ exists on $[a,b]$, let $$A_n=\frac{b-a}{n}\sum_{k=1}^{n}f\bigg(a+\frac{k(b-a)}{n}\bigg) -\int_{a}^{b}f(x)\mathrm{d}x,$$ then $$\color{red}{\lim_{n\to \infty}nA_n=\frac{f(b)-f(a)}{2}(b-a).}$$

Answer 2

Assuming $p > -1$, continuing your split up series.

1 .This is was WA gives us for the first summation. First sum, e.g. $p=3$ $$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\left(\frac{k}{n}\right)^p\ln \frac{k}{n} = -\frac{1}{(p+1)^2}$$. EDIT: Using the hint from @Riemann, this sum is equal to $\int_0^1x^p\ln(x)dx=-\frac{1}{(p+1)^2}$, for $p>-1$.

2. Second $$\frac{\ln n}{n}\sum_{k=1}^n\left(\frac{k}{n}\right)^p - \frac{\ln n}{p+1}=\ln n\left[\frac{1}{n}\sum_{k=1}^n\left(\frac{k}{n}\right)^p-\frac{1}{p+1}\right] = \ln n\left[\frac{1}{n^{p+1}}\sum_{k=1}^nk^p-\frac{1}{p+1}\right].$$ $\sum_{k=1}^nk^p$ can be written using Bernoulli numbers as $$\sum_{k=1}^nk^p=\frac{n^{p+1}}{p+1}+\frac{1}{2}n^p+\sum_{k=2}^p\frac{B_k}{k!}p^\underline{k-1}n^{p-k+1},$$ and thus we obtain $$\frac{1}{p+1}+\frac{1}{2}\frac{n^p}{n^{p+1}}+\frac{n^{p+1}}{n^{p+1}}\sum_{k=2}^p\frac{B_k}{k!}p^\underline{k-1}n^{-k}-\frac{1}{p+1}=$$ $$\frac{1}{2n}+\sum_{k=2}^p\frac{B_k}{k!}p^\underline{k-1}n^{-k}.$$ And therefore, since $\lim_{n\to\infty}\frac{\ln n}{n} = 0$, $$\lim_{n\to\infty}\frac{\ln n}{2n}+\sum_{k=2}^p\frac{B_k}{k!}p^\underline{k-1}\frac{\ln n}{n^{k}}=0$$

Combing the two series, $$\lim_{n\to\infty}\ln a_n = -\frac{1}{(p+1)^2} + 0$$ and (to me somehow surprising) $$\lim_{n\to\infty}a_n=e^{-\frac{1}{(p+1)^2}}$$