Question

Find the sum of the following series $\frac{x^3}{\lfloor 3}+\frac{x^5}{\lfloor 5}+\frac{x^7}{\lfloor 7}+....$

Answer 1

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

$e^{-x}=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+....$

$e^x-e^{-x}=2x+2S$

$S=\frac{e^x-e^{-x}-2x}{2}$