How Can We Prove This Exponential Identity?

[muzialis]

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

 

[Erland]

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

There is a typo in the middle formula. It should be: $\lim_{N \to \infty} \sum_{r=0}^{N} \frac {N!}{r! (N-r)!} (\frac{y}{N})^{r}$.

It is then easy to see that this is the binomial expansion of the right side.
One also sees that for each fixed $r$, the $r$-term in the middle sum tends to $y^r/r!$ as $N \to\infty$.

It remains to convince oneself that everything works out with the limits.

 

[muzialis]

Erland,
many thanks for pointing this out, I am a little bit closer now to understanding, many thanks
!