If $\cosh (x-\log 3)=\sinh x$,then $x=$

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:

$1 + \frac{3}{1!} + \frac{5}{2!} + \frac{7}{3!} + ....\infty = $

If ${T_n} = \frac{{{3^n}}}{{2(n!)}} - \frac{1}{{2(n!)}}$,then ${S_\infty } = $

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:

$1 + x \log_e a + \frac{x^2}{2!} (\log_e a)^2 + \frac{x^3}{3!} (\log_e a)^3 + \dots = $