How do you prove or disprove a 1:k correspondance between two series that converge to the same thing?

Question

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.

Answer 1

No, you can't assume that. I think the important point is that having the sum of the series be something is only one constraint. There are infinitely many terms, so you can perturb lots of them in lots of ways without changing the sum.

Say you have a very well behaved series like the classic one for $e, \sum_{i=0}^\infty \frac 1{i!}$. This converges very rapidly. It doesn't take many terms to get very close to $e$. Now I will make a new series by adding $\frac {1,000,000}n$ to every tenth term and subtracting that from the term after. The series still converges, though conditionally now, to $e$ and is Cauchy, but it takes lots more terms before you stay within a given $\epsilon$ of $e$

Answer 2

It may be simpler to rephrase your question in terms of convergent sequences instead of series (since a series is the limit of a sequence of partial sums).

So essentially you have $a_n \to a$ and $b_m \to a$, and you want to know if both sequences have a common subsequence; that is, you want to know if there are subsequences $(a_{n_k})_k$ and $(b_{m_k})_k$ such that $a_{n_k} = b_{m_k}$ for all $k$.

I'm not sure if there is a strategy to answer this question in generality; it would have to depend on what the sequences (or series) are.