Quotient of infinite products

Question

I have a very simple question, but I never learned about infinite products and now have to use them.

Am I right in assuming that $$ {\prod_{k=1}^{\infty} f(k)\over\prod_{k=1}^{\infty} g(k)}=\prod_{k=1}^{\infty} {f(k) \over g(k)} $$

even when both products of $f(k)$ and $g(k)$ don't converge? Can someone stear me in the right direction?

Answer 1

this doesnt work for infinite products in general, as karakusc pointed out...
but it does work for finite one's: $$ {\prod_{k=1}^{N} f(k)\over\prod_{k=1}^{N} g(k)}=\prod_{k=1}^{N} {f(k) \over g(k)} $$ and: $$ \lim_{N\to \infty}{\prod_{k=1}^{N} f(k)\over\prod_{k=1}^{N} g(k)}=\lim_{N\to \infty}\prod_{k=1}^{N} {f(k) \over g(k)} $$

Answer 2

This is not true in general. Consider $f(k)=g(k)=(-1)^k$. The left-hand side does not converge, but the right hand side is 1.

Answer 3

In the case that one of the two infinite products is not convergent, the left-hand-side is not defined.

Moreover, suppose $f(k), g(k) >0$. Then passing to logarithms you get

$$\sum_{k=1}^{+\infty} \log f(k) - \sum_{k=1}^{+\infty} \log g(k) = \sum_{k=1}^{+\infty} ( \log f(k)- \log g(k))$$

If we suppose that the right-hand-side is well defined, is this correct? The answer is: in the case we are dealing with non convergent series, the left-hand-side is not defined (for example you have the indeterminate form $+ \infty - \infty$).

What we can say is : supposing the two infinite products both converge, then your equality holds.