1. Let's clarify the problem: You are asking if there is a theorem stating that two sequences or series converge together. 2. One important theorem related to convergence of sequences is the \textbf{Comparison Test} for series, which states that if $0 \leq a_n \leq b_n$ for all $n$ beyond some index, and if $\sum b_n$ converges, then $\sum a_n$ also converges. This implies they converge together in a certain sense. 3. Another relevant result is the \textbf{Limit Comparison Test}, which says if $\lim_{n \to \infty} \frac{a_n}{b_n} = c$ where $c$ is a finite positive number, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge. 4. For sequences, if $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} b_n = L$, then both sequences converge to the same limit $L$. 5. These theorems help us understand when two sequences or series converge together by comparing their terms or limits. 6. In summary, yes, there are theorems like the Limit Comparison Test and Comparison Test that state conditions under which two series converge together.