- #1
muzialis
- 166
- 1
Hi All,
I am struggling to prove the following identity
$$ 1 + y + \frac{1}{2!}y^2 + \frac{1}{3!}y^3 + \dots = lim_{N \to \infty} \sum_{r=0}^{N} \frac {N!}{r! (N-r)!} \frac{y}{N}^{r} = lim_{N \to \infty} (1 + \frac{y}{N})^N $$
any hint would the most appreciated. I understand the left-most term is the Taylor series for the exponential function, and the right-most term is also used as a definition of such function, yet I would like to know how the two are explicitly shown to be equivalent.
Thanks as usual
I am struggling to prove the following identity
$$ 1 + y + \frac{1}{2!}y^2 + \frac{1}{3!}y^3 + \dots = lim_{N \to \infty} \sum_{r=0}^{N} \frac {N!}{r! (N-r)!} \frac{y}{N}^{r} = lim_{N \to \infty} (1 + \frac{y}{N})^N $$
any hint would the most appreciated. I understand the left-most term is the Taylor series for the exponential function, and the right-most term is also used as a definition of such function, yet I would like to know how the two are explicitly shown to be equivalent.
Thanks as usual