Consider the follow expression: $$\displaystyle{\frac{n \# \cdot n!}{p_{n}\#}}, n \in \mathbb{Z^*}$$
In this thread, the notation will mean the following:
- $n\#$ is the primorial of $n$, defined as being the product of all prime numbers less than and including $n$ if $n$ itself is prime
- $n!$ is the factorial of $n$, defined as being the product of all positive integers equal to and less than $n$
- $p_n\#$ is the priomorial function, defined as being the product of the first $n$ prime numbers
- $\mathbb{Z^*}$ is the set of non-negative integers
For example, let's consider $n=6$: $$\displaystyle{\frac{6 \# \cdot 6!}{p_{6}\#}} = \displaystyle{\frac{(5 \cdot 3 \cdot 2) \cdot (6 \cdot 5 \cdot 4 \cdot 3 \cdot 2)}{2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13}} =\frac{720}{1001} \approx 0.71928$$
Based off this information, how would one go about evaluating $\displaystyle{\lim_{n \to \infty}} \frac{n \# \cdot n!}{p_{n}\#}$? Additionally, will the value of the limit be the same in this scenario: $\displaystyle{\lim_{n \to \infty}} \frac{n \# \cdot n!}{p_{n-1}\#}$?
Continuing on, if we were to define function $f$ such that $f(x)=\displaystyle{\frac{x \# \cdot x!}{p_{x}\#}}$ and assume that the curve of best fit is being mapped to each fixed positive integer value of $n$ including $0$. That is to say, the curve must pass through $(0, f(0)), (1, f(1)), (2, f(2)), \text{ etc}$.
What is the best estimation for the area enclosed by said curve and the $x$-axis from $x=0$ to $x=\infty$? I.e:
$$\int_0^\infty \displaystyle{\frac{x \# \cdot x!}{p_{x}\#}} dx$$