Engineering Mathematics: Useful Exponential Identity

UEI.ID.1: Series

$(1+x)\,e^{x} = 1 + \sum _{n=0}^{\infty }(\frac{n+1}{n!})\,x^{n}$

Proof

$(1+x)\,e^{x} = e^{x}+ x\,e^{x}$

(1)

$e^{x}+ x\,e^{x} = \sum _{n=0}^{\infty }\frac{x^{n}}{n!} + \sum _{n=0}^{\infty }\frac{x^{n+1}}{n!}$

(2)

$e^{x}+ x\,e^{x} = 1+ \sum _{n=1}^{\infty }\frac{x^{n}}{n!} + \sum _{n=0}^{\infty }\frac{x^{n}}{(n-1)!}$

(3)

$e^{x}+ x\,e^{x} = 1 + \sum _{n=0}^{\infty }(\frac{1}{n!}+\frac{1}{(n-1)!})\,x^{n}$

(4)

$e^{x}+ x\,e^{x} = 1 + \sum _{n=0}^{\infty }(\frac{n+1}{n!})\,x^{n}$

(5)

$(1+x)\,e^{x} = 1 + \sum _{n=0}^{\infty }(\frac{n+1}{n!})\,x^{n}$

Engineering Mathematics