91Ó°ÊÓ

Partial sums are a crucial concept in understanding series. They help us determine whether a series is convergent or divergent. Imagine you have an infinite series represented as \(\Sigma a_n\), which consists of terms \(a_1, a_2, a_3, \ldots\). A partial sum \(s_n\) is the sum of the first \(n\) terms: \(s_n = a_1 + a_2 + \cdots + a_n\). Each partial sum provides a snapshot of the cumulative total of the series at that point.

The series is said to be convergent if these partial sums approach a finite number as \(n\) becomes very large. If they do not settle on a finite number, then the series is divergent. In the given problem, knowing that \(s_n \leq 1000\) helps us conclude something important about convergence. It hints that these sums will not grow beyond this boundary, suggesting the potential for convergence.

A bounded sequence is a sequence of numbers that stays within a specific range and does not go off to infinity or negative infinity. In mathematical terms, a sequence \(s_n\) is bounded if there exists a number \(M\) such that \(|s_n| \leq M\) for all \(n\).

In the original exercise, the series' partial sums \(s_n\) were given to satisfy \(s_n \leq 1000\). This means that the sequence \(s_n\) of partial sums is bounded above by 1000. Boundedness is a key indicator of how series behave, especially when determining convergence. If a sequence has no bounds, it can become infinite, and whether it converges will be uncertain.

A monotonic sequence is a sequence that is entirely non-decreasing or non-increasing. For example, a sequence \(s_n\) is non-decreasing if each term is greater than or equal to the previous term: \(s_n \leq s_{n+1}\) for all \(n\). Conversely, it is non-increasing if \(s_n \geq s_{n+1}\).

In this particular problem, since the terms \(a_n\) of the series are positive, the partial sums \(s_n\) are also forming a non-decreasing sequence. When a sequence is both monotonic and bounded, it hints strongly at convergence.

• Positive terms ensure every subsequent sum doesn't decrease.
• Boundedness prevents the sequence from escaping to infinity.

This is why monotonicity, coupled with boundedness, is a compelling argument for convergence.

A convergent series is one where the sum of its infinite terms approaches a specific, finite value. When analyzing such a series, the behavior of its partial sums is of primary interest.

For a series \( \Sigma a_n \) to converge, its sequence of partial sums \( s_n \) must approach a finite limit as \( n \to \infty \). This exercise demonstrates a critical aspect of this process: If the partial sums are both bounded and monotonic, they must converge by the Bounded Monotonic Sequence Theorem.

• The theorem asserts that any sequence that is both bounded and monotonic will converge.
• Hence, \( s_n \) converges to a finite limit, proving \( \Sigma a_n \) is convergent.

Understanding convergence through these properties allows us to grasp why the exercise concludes with a confirmed convergence of the series.