Let $S = \frac{2}{1} {}^{n}C_{0} + \frac{2^{2}}{2} {}^{n}C_{1} + \frac{2^{3}}{3} {}^{n}C_{2} + \ldots + \frac{2^{n+1}}{n+1} {}^{n}C_{n}$. Then, $S$ equals

Let $[ \cdot ]$ denote the greatest integer function and $f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^n \left[ \frac{k^2}{3^x} \right]$. Then $12 \sum_{j=1}^{\infty} f(j)$ is equal to ........... .

$\mathop {\lim }\limits_{n \to \infty } {\left\{ {\left( {1 + \frac{{{1^2}}}{{{n^2}}}} \right)\left( {1 + \frac{{{2^2}}}{{{n^2}}}} \right)\left( {1 + \frac{{{3^2}}}{{{n^2}}}} \right) \dots \left( {1 + \frac{{{{(n - 1)}^2}}}{{{n^2}}}} \right)} \right\}^{1/n}}$ equals to:

$\lim _{n \rightarrow \infty} \frac{(2n(2n-1) \dots (n+1))^{1/n}}{n} = $

For a positive integer $n$,define $f(n) = n + \sum_{r=1}^n \frac{16r + (9-4r)n - 3n^2}{4rn + 3n^2}$. Then,the value of $\lim_{n \rightarrow \infty} f(n)$ is equal to

If $\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{4 r^3}{r^4+n^4}=p$, then $e^p=$