S=1+2/2!+3/3!+...+n/n!+...=1+1/1!+1/2!+...+1/(n-1)!+...
Since the series is infinite, S=∑1/n! which is the series for e (base of natural logarithms).
ex=1+x+x2/2!+...+xn/n!+...
d(ex)/dx=ex by definition, and dS/dx=1+x+...+xn-1/(n-1)!+...
When x=1, this series is the same as S, so S=e1=e.