Suppose I have two completely different series representations of a function that can't be conventionally manipulated into each other, but converge to the same function none-the-less, like a hypergoemetric series and a taylor series (there may be a way to convert those into each other but for now assume there isn't).
Now, even though both series converge the same function at infinite terms, I don't know if there is any proof to assume these two series would also converge to the same partial sum in the event one does not take the sums to infinity. Is there? Or how do I prove/disprove that notion for any given pair of series?
Because there doesn't necessarily have to be a 1:1 correspondence. It make take 3 or 4 partial terms of one series just to equal the second partial term of the other series, but the goal is to prove whether such a proportionality exists at all before trying to find it.