91Ó°ÊÓ

In calculus, understanding the concept of convergence of a sequence is a foundational step. A sequence \(\{s_n\}\) is said to converge to a limit \(L\) if, as the sequence progresses (i.e., as \(n\) becomes larger and larger), the terms \(s_n\) get closer and closer to \(L\). The formal definition states that a sequence converges to \(L\) if for every positive number \(\epsilon > 0\), no matter how small, there exists a positive integer \(N\) such that for all \(n > N\), the absolute difference between \(s_n\) and \(L\) is less than \(\epsilon\).

Proving convergence involves showing that the chosen \(N\) works for the stipulated \(\epsilon\). Let's apply this with an example where \(\epsilon\) is set as \(\frac{L}{2}\) since \(L > 0\). This value of \(\epsilon\) is meaningful because it directly relates to the known positive limit \(L\). By the definition of convergence, there must be an \(N\) where for all \(n > N\), \(\left|s_n - L\right| < \frac{L}{2}\). This creates an interval \(\left(\frac{L}{2}, \frac{3L}{2}\right)\) that \(s_n\) will fall into. Clearly, if \(s_n\) is within this interval, \(s_n\) must be greater than zero for \(n > N\), thus substantiating the initial claim and proving the convergence of the sequence to the positive limit \(L\).

To identify whether a sequence converges, certain criteria must be satisfied. One fundamental criterion is that the sequence must be bounded after some point—meaning there exists some \(N\) such that all subsequent \(s_n\) are within a specified range. A second essential criterion is that the sequence becomes arbitrarily close to the limit \(L\), a property reflected in the formal definition, which states that the difference between the sequence terms and the limit must be less than any positive \(\epsilon\), no matter how small, after a certain point in the sequence. This is why in convergence proofs, choosing an \(\epsilon\) that is related to the limit—like \(\frac{L}{2}\) in our exercise—is effective for illustrating that the terms of the sequence are indeed converging to the limit \(L\).