91Ó°ÊÓ

When we discuss the **limit of a sequence**, we are trying to understand what happens to the values of a sequence as we increase the index value, often denoted by \( n \). A sequence converges if the terms become closer and closer to a specific number as \( n \) tends towards infinity. More formally, a sequence \( \{a_n\} \) is said to converge to \( L \) if for every positive number \( \epsilon \), there exists a positive integer \( N \) such that for all \( n > N \), the absolute difference \( |a_n - L| < \epsilon \). This means that beyond a certain index, all terms of the sequence are arbitrarily close to \( L \).
For the sequence given by \( a_n = \frac{n^2 - 1}{2n} \), evaluating its limit helps determine whether it converges or diverges. Upon simplification, you find that the dominant term as \( n \to \infty \) is \( \frac{n}{2} \), which grows without bound, indicating that the limit does not exist in the finite realm, and thus the sequence does not converge.

Understanding the **asymptotic behavior** of a sequence helps in comprehending how the sequences behave as they stretch towards infinity. It's a way to describe the limiting behavior of a sequence without necessarily finding an exact limit.
In the exercise, to identify the asymptotic behavior, the sequence \( \frac{n^2 - 1}{2n} \) was expressed as \( \frac{n}{2} - \frac{1}{2n} \). Here, \( \frac{n}{2} \) suggests that as \( n \) increases, the sequence largely behaves like \( n/2 \), growing larger and larger. The term \( \frac{1}{2n} \) becomes negligible as \( n \) grows very large, having less and less impact on the total value of the sequence. Thus, asymptotically, the sequence behaves like \( \frac{n}{2} \).
This insight makes it clear that as \( n \to \infty \), the sequence's terms also move towards infinity, confirming that it does not settle at a fixed value, which reflects divergence.

An **infinite sequence** refers to a list of numbers in a specific order that continues indefinitely. In mathematical terms, it means there is no last term. Some infinite sequences may "settle down" to a particular number, implying they converge, while others do not and are said to diverge.
The infinite sequence in question, \( \left\{\frac{n^2 - 1}{2n}\right\} \), does not become stable as \( n \to \infty \). Instead, its terms grow bigger and approach infinity. This behavior characterizes diverging infinite sequences.
Each new term of this sequence is derived by plugging in values of \( n \) that continue to increase without bound. Because the main term \( \frac{n}{2} \) dominates, effectively leading to an infinite progression, the sequence lacks any fixed limit. This is crucial for differentiating between sequences that may have trivial patterns but never actually converge to a finite point.