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

$1 + \frac{a - bx}{1!} + \frac{(a - bx)^2}{2!} + \frac{(a - bx)^3}{3!} + \dots \infty = $

$\frac{1}{2!} + \frac{1+2}{3!} + \frac{1+2+3}{4!} + \ldots$ is equal to :

If $a = \sum\limits_{n = 0}^\infty {\frac{{{x^{3n}}}}{{(3n)!}}} ,\,b = \sum\limits_{n = 1}^\infty {\frac{{{x^{3n - 2}}}}{{(3n - 2)!}}} $ and $c = \sum\limits_{n = 1}^\infty {\frac{{{x^{3n - 1}}}}{{(3n - 1)!}}} $ then the value of ${a^3} + {b^3} + {c^3} - 3abc = $

If $2 \sinh x = \cosh x$,then $x =$

The sum of the series $\sum_{n=1}^{\infty} \frac{n^{2}+6 n+10}{(2 n+1) !}$ is equal to :