Question about uniqueness function series Please pardon me if this is elementary, but I've looked hard for an answer to this and am very surprised I have yet to find a good one. My question is simple: under what condition(s) does $$\sum_{n=1}^{\infty}f_n(x) = \sum_{n=1}^{\infty}g_n(x)$$ imply that $$f_n(x) = g_n(x)$$?
 A: It's hard to find any reasonable conditions for such an implication. The reason is simply that given any series of the form $\sum_{n=1}^\infty f_n(x)$ that converges, then changing the order of summation, or making small or large changes to $f_1$ only to absorb these in other $f_k(x)$ gives too much freedom for finding infinitely many possibilities for functions $g_k(x)$ that will add up to the same value. 
If you place some severe restrictions on the functions though then some (not terribly interesting) implications are possible. For instance, if all functions attain non-negative values, and moreover the inequalities $f_k(x)\le g_k(x)$ hold for all $x$ in the relevant domain, then equality of the series will imply the equality of the functions (in the relevant domain). 
But really, this is hopeless. It's like asking for conditions that will assure that two series of real numbers are equal. There are simply too many possibilities. Even with sums of just two real numbers, what conditions are there on $a,b,c,d$ that ensure that if $a+b=c+d$, then $a=c$ and $b=d$??? 
