Suppose $A=\sum_{k=0}^{\infty}f(k)=B\prod_{n=0}^{\infty}g(n)$. If $A$ and $f(k)$ are known, how to find $B$ and $g(n)$?

Question

My question like "some-to-product" or vice versa.

See the following example (for reference, see here and here);

$$\pi=\sum_{k=0}^{\infty}\frac{4(-1)^k}{2k+1}=2\prod_{n=0}^{\infty}\frac{4n^2+8n+4}{4n^2+8n+3}$$

My question is:

If

$$A=\sum_{k=0}^{\infty}f(k)=B\prod_{n=0}^{\infty}g(n)$$

Is there a way/procedure one can use to find out $B$ and $g(n)$ if $A$ and $f(k)$ are given, and $B \ne A$?

So if $A=\pi$ and $f(k)=\frac{4(-1)^k}{2k+1}$, then one can find (using that way, if any) that $B=2$ and $g(n)=\frac{4n^2+8n+4}{4n^2+8n+3}$.

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EDIT:

I know that $g(n)$ will change as $B$ changes. Hence infinitely many combinations of $B$ and $g(n)$ are there. To understand me better, say we choose a real $B$, (then) we find out the (suitble) $g(n)$.

NOTE:

Some people commented/answered with defining $g(0)$ and $g(n)$ for $n>=1$ separately!

This is not matching my question above, $\color{red}{A=\sum_{k=0}^{\infty}f(k)=B\prod_{n=0}^{\infty}g(n)}$.

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Hopefully my question is clear, and hopefully there is a way.

Your help would be appreciated. THANKS!

Answer 1

They aren't uniquely defined. You can take e.g. $B = 1$, $g(0) = A$, $g(i) = 1$ for $i > 0$, or many, many other combinations.