The value of $\sum\limits_{n = 1}^\infty {\frac{{^n{C_0} + ... + ^n{C_n}}}{{^n{P_n}}}} $ is

In the expansion of $(1 + x + x^2)e^{-x}$,the coefficient of $x^2$ is

Let $S_{n} = 1 \cdot (n-1) + 2 \cdot (n-2) + 3 \cdot (n-3) + \dots + (n-1) \cdot 1$,for $n \geq 4$. The sum $\sum_{n=4}^{\infty} \left( \frac{2 S_{n}}{n!} - \frac{1}{(n-2)!} \right)$ is equal to:

Let $C(\theta) = \sum_{n=0}^{\infty} \frac{\cos(n\theta)}{n!}$. Which of the following statements is $FALSE$?

$\frac{1 + \frac{2^2}{2!} + \frac{2^4}{3!} + \frac{2^6}{4!} + \dots \infty}{1 + \frac{1}{2!} + \frac{2}{3!} + \frac{2^2}{4!} + \dots \infty} = $

$\sum_{n=1}^{\infty} \frac{2n}{(2n+1)!}$ is equal to