Is it possible by elementary means to derive a bound on the modulus of $z $ for which we have that
$$\left(1+\frac{|z|}{n+1}+\frac{|z|^2}{(n+1)(n+2)}+\frac{|z|^3}{(n+1)(n+2)(n+3)} \dots\right)\le 2$$?
By elementary means I mean methods from the usual exposition of (comples) series. If you should use it I may also say that I take the exponential function $\exp(z) $ to be defined by the convergent power series $\sum_{n=0 } ^{\infty } \frac {z^n } {n!} $.
Much grateful!