91Ó°ÊÓ

The limit of a sequence is a fundamental concept in calculus. When we say that \( \lim a_n = \infty \), we are stating that the sequence \( a_n \) increases without bound as \( n \) becomes larger. This means that for any substantial positive number \( M \), no matter how large, there is a point in the sequence beyond which all terms exceed \( M \).
This property is crucial for understanding why certain series diverge. If the terms of our sequence \( a_n \) are positive and individually grow larger and larger, there is no ceiling limitation for the collective sum of these terms. This lack of restriction sets up the basis for our conclusion regarding divergence in such scenarios.

A positive term series implies that each term in the series is a positive number. This is significant because the sum of positive numbers can only grow or remain constant; it can never decrease.
In our context, given that each \( a_n > 0 \), the sum \( \Sigma a_n \) cannot converge if \( a_n \) itself becomes indefinitely large. This is because a converging series suggests that its sum approaches a specific finite number as more terms are added. However, when all terms are positive and increasingly large, the sum naturally trends toward infinity, which indicates divergence. The infinite growth of terms ensures that the series sum cannot "settle" at any particular finite value.

The idea of an infinite limit is pivotal when discussing series divergence. It refers to situations where sequence terms increase perpetually without boundary. In mathematical terms, if the limit of a sequence's terms is infinite, the terms do not approach any finite number; instead, they continue to grow.
This behavior directly opposes the necessary condition for convergence in series. For a series to converge, its sequence terms need to diminish as they progress toward the end. The requirement is that \( \lim_{n \to \infty} a_n = 0 \) for convergence to be possible if each \( a_n \) is positive. In our scenario, since \( \lim a_n = \infty \), the terms do not shrink; they magnify instead. Consequently, the sum of such terms, \( \Sigma a_n \), cannot converge, and thus diverges.