The square of a infinite series. Say we have $\sum_{n=1}^{\infty} f(n,x)=f(x)$, which often happens with Taylor series:
Can we express:
$$\left(\sum_{n=1}^{\infty} f(n,x) \right)^2$$
As something that does not involve the square. I.e  can multiply this out :
$$\left(f(1,x)+f(2,x)+f(4,x)+.... \right)\left(f(1,x)+f(2,x)+f(3,x)+...\right)$$
I know by distributive property we have:
$$=f(1,x)\sum_{n=1}^{\infty} f(n,x)+f(2,x)\sum_{n=1}^{\infty} f(n,x)+f(3,x)\sum_{n=1}^{\infty} f(n,x)+..$$
Can we simplify further?
Why I ask this is that I'm really interested if we can express
$\left(\frac{\ln (x)}{x-1} \right)^2$ as a series by only using the Taylor series for $\ln x$. I know I can just write the coefficients and multiply them out , but the new coefficients to the new series do not form an easy pattern to right down.
 A: What you are referring to is called the Cauchy product. If you consider multiplying out term by term, you might expect to see something like
$$(a_1+a_2+a_3+\cdots)(b_1+b_2+b_3+\cdots)=a_1b_1+(a_1b_2+a_2b_1)+(a_1b_3+a_2b_2+a_3b_1)+\cdots$$
where we are generalizing the FOIL method in the only way that makes sense.
The question is $(1)$ whether the sum converges and $(2)$ if it converges to what we expect it to (that is, if the first converges to $A$ and the second to $B$, then the product converges to $AB$). There is no answer in general but there are a few theorems to keep in mind:
$(1)$: if one (or both) of the series converges absolutely (and the other converges), then the product converges to $AB$.
$(2)$ if all three series converge, then the product indeed converges to $AB$.
In any particular case, like the one you mentioned, it may be difficult to show convergence of the product since the terms do not generalize well. One possibility is to find a bound for the product (perhaps using partial fractions) and show that the bound goes to zero face enough to guarantee convergence.
