TITLE: Change from product to sum
QUESTION [7 upvotes]: We know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$
That's how we convert from a product to a sum.
So what happens if we go a little further?
That is : $$\prod\limits_{a}^{b} {f(x)} = \sum\limits_{?}^{?} {?}$$
$$\text{where } b>a$$

What will be in the place of those $'?'$
So if $f(x) = \sin{(x)} \text{ or } = x^3 \text{ or } = a^x$ or something like that?
Or what if $b=\infty$ ?

I guess this is a silly question but please help! Thanks!

REPLY [3 votes]: If you don't mind a high tech answer, here's a consequence of the binomial theorem:
\begin{align}
\prod_{i=1}^N f_i &= \frac1{2^{N-1}N!}\sum_{\sigma_{2,\dots,N} = \pm} \left(\prod_{i=2}^N \sigma_i \right) \left(f_1+\sum_{i=2}^N \sigma_i f_i\right)^N\\
\prod_{i=1}^N f_i &= \frac1{2^{N-1}N!}\sum_{\sigma_{2,\dots,N} = \pm} \left(\prod_{i=2}^{N} \sigma_i \right) \left(f_1+\sigma_{N+1} f_{N+1} +\sum_{i=2}^{N} \sigma_i f_i\right)^N + f_{N+1}\prod_{i=1}^{N-1} f_i,
\end{align}