Question

Find the sum of the following series $1+\frac{2x^2}{\lfloor 2}-\frac{x^3}{\lfloor 3}+\frac{7x^4}{\lfloor 4}-\frac{23x^5}{\lfloor 5}+\frac{121x^6}{\lfloor 6}-....$

Answer 1

$\Rightarrow S=1+(1-1)x+\frac{1+1}{2}{x}^2-\frac{2!-1}{3!}{x}^3+\frac{3!+1}{4!}{x}^4+....$

$=(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+....)-(x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}+....)$

$S=e^x-\log (1+x)$