How to go from a sum to a product and a product to a sum?

Question

I have read here (third post down) that exponentials turn sums into products and logarithms turn products into sums. Can someone please further explain this?

Answer 1

Consider the set $\mathbb R$ of all real numbers together with the operation of addition and consider the set $\mathbb R_+$ of all positive real numbers with the operation of multiplication. Both of these are examples of groups. The properties that $e^{a+b}=e^a\cdot e^b$ and $\ln(ab)=\ln(a)+\ln(b)$, show that $\exp:(\mathbb R,+)\to (\mathbb R_+,\cdot)$ is a group isomorphism. Its inverse is $\ln$.

Answer 2

The intuition of course comes from the identities $e^{a+b} = e^a e^b$ and $\log{xy} = \log{x} + \log{y}$. However, this ignores issues of convergence. There are a couple nice theorems that link infinite products and infinite series though:

Theorem 1 If $a_k > -1$ for all $k$, then the infinite product $\Pi_{k=1}^\infty (1+a_k)$ converges if and only if the infinite series $\sum\limits_{k=1}^\infty \log{(1+a_k)}$ converges.

Theorem 2 If $\sum\limits_{k=1}^\infty a_k^2 < \infty$, then the product $\Pi_{k=1}^\infty (1+a_k)$ converges if and only if the series $\sum\limits_{k=1}^\infty a_k$ converges.

I picked these out of Peter Duren's book Invitation to Classical Analysis. The proof of theorem 1 is pretty simple, and theorem 2 follows from theorem 1 and the inequality $$\frac{1}{3} x^2 < x - \log{(1+x)} < x^2 \; , \qquad |x|<\delta \; ,$$ for sufficiently small $\delta$.