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

The sum of the series $\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \dots$ is

The value of the infinite series $\frac{1^{2}+2^{2}}{3 !} + \frac{1^{2}+2^{2}+3^{2}}{4 !} + \frac{1^{2}+2^{2}+3^{2}+4^{2}}{5 !} + \dots$ is:

The sum of the series $\frac{1}{1 \times 2} + \frac{1 \times 3}{1 \times 2 \times 3 \times 4} + \frac{1 \times 3 \times 5}{1 \times 2 \times 3 \times 4 \times 5 \times 6} + \dots \infty$ is

The sum of the series $\frac{1^2}{1 \cdot 2!} + \frac{1^2 + 2^2}{2 \cdot 3!} + \frac{1^2 + 2^2 + 3^2}{3 \cdot 4!} + \dots + \frac{1^2 + 2^2 + \dots + n^2}{n(n + 1)!} + \dots \infty$ is equal to:

If $S_n$ denotes the sum of the products of the first $n$ natural numbers taken two at a time,then $\sum\limits_{n = 0}^\infty \frac{S_n}{(n + 1)!} = $