Question

Find the sum of the following series $1+2x+\frac{2^2-1}{\lfloor 2}\cdot \frac{x^2}{1}+\frac{3^2-1}{\lfloor 3}\cdot \frac{x^3}{2}+\frac{4^2-1}{\lfloor 4}\cdot \frac{x^4}{3}+.....$

Answer 1

$\Rightarrow S=1+2x+\frac{3}{2!}{x}^2+\frac{4}{3!}{x}^3+\frac{5}{4!}{x}^4+.....$

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

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

$=e^x+x{e}^x$