I want to show that for every natural $n$, the quotient
$$\frac{(b)_n}{b^n} \to 1 \quad \mbox{as} \quad b\to \infty$$
Where $(b)_n = b(b+1)...(b+n-1)$ is the rising factorial. This is so that I can show that:
$$\lim_{b\to\infty}\sum_{n=0}^{\infty} \frac{(b)_n}{b^n} \frac{x^n}{n!} = e^x.$$
I know that the series is absolutely convergent so I can move the limit inside. But how do I show that the quotient tends to $1$? It seems intuitively obvious, since for very large $b$, each of the terms $\frac{b+k}{b}$ is nearly one, but how do I formally show this?
When b tends to infinity, n is constant, so this is the limit of the product of n terms, each of which tends to 1.
$\lim \frac{b+k}{b} = 1$ so $\lim \prod \frac{b+k}{b} = \prod \lim \frac{b+k}{b} = 1$