In a lot of older texts I see a lot of constants and functions represented by infinite products like $$ \frac{a}{a-z}= \prod_{n=1}^{ \infty}e^{ \frac{1}{n}( \frac{c}{z})^{n}},$$
but nowadays all I see are infinite summations, usually in the form of a Taylor series.
Is there a way to convert an infinite sum like a Taylor series into an infinite product and is there any practical application for it?
Note that the summation can be converted into a product using the fact that $$e^a\cdot e^b = e^{a+b},$$ and $$e^{\ln(x)} = x.$$ For example, by Taylor series, $$\ln\left(\frac{a}{a-z}\right) = \sum_{n=1}^{\infty} \frac{1}{n}\cdot \left(\frac{a}{z}\right)^n.$$ Therefore, $$\frac{a}{a-z} = e^{\ln\left(\frac{a}{a-z}\right)} = e^{\sum_{n=1}^{\infty} \frac{1}{n}\cdot \left(\frac{a}{z}\right)^n} = \prod_{n=1}^{\infty} e^{\frac{1}{n}\cdot \left(\frac{a}{z}\right)^n}.$$