I have a problem with finding the limit of the following sequence:
$$\lim\limits_{n \to\infty}\frac{\prod_{k=1}^n{(1+x^k)}}{(1+(nx)^n)^{\frac{n+1}{2}}}.$$
I derived the sequence to the following form: $$\lim\limits_{n \to\infty}\frac{{(1+x)(1+x^2)\ldots(1+x^n)}}{n^{\frac{n(n+1)}{2}}(\frac{1}{n^n}+x^n)^{\frac{n+1}{2}}},$$ but still can't manage to find the limit.