Final answer:
For a sequence with positive terms where the limit of the ratio of successive terms is 2023, one can show that the sequence is unbounded and increases monotonically. Since it's not bound, the limit of the terms must approach infinity due to the nature of the ratio approaching a large finite number.
Step-by-step explanation:
Let \((a_n)_{n=1}^{\infty}\) be a sequence with positive terms such that \(\lim_{n\to\infty}\frac{a_n}{a_{n+1}}=2023\). By the definition of limit, for every \(\epsilon > 0\), there exists an integer \(N\) such that \(|\frac{a_n}{a_{n+1}} - 2023| < \epsilon\) for all \(n > N\). However, as \(a_n\) and \(a_{n+1}\) are both positive, we can deduce that \(a_{n+1} \geq a_n\) eventually. If we suppose that \((a_n)_{n=1}^{\infty}\) is bounded above by some \(M\), it would contradict the fact that \(a_{n+1} / a_n\) approaches 2023 since the sequence would not have enough room to grow to sustain the ratio, hence the sequence has to be unbounded. Applying the Algebraic Limit Theorem to \(a_{n+1}/a_n\), and knowing the sequence is unbounded and monotonically increasing past a certain point, we conclude that \(\lim_{n\to\infty}a_n = \infty\).