Problem 55 - Essential Calculus Early Transcendentals

Question

Suppose that a series \(\Sigma a_{n}\) has positive terms and its partial sums \(s_{n}\) satisfy the inequality \(s_{n} \leqslant 1000\) for all \(n .\) Explain why \(\Sigma a_{n}\) must be convergent.

Step-by-Step Solution

Answer

The series converges because its partial sums are bounded.

Step 1: Understanding the Problem

We need to determine why a series with positive terms remains convergent if its partial sums are bounded by a constant value, in this case, 1000.

Step 2: Defining Series and Partial Sums

The series \( \Sigma a_{n} \) is the sum of its terms \( a_1 + a_2 + a_3 + \ldots \). The \( n \)-th partial sum of the series, denoted as \( s_n \), is \( s_n = a_1 + a_2 + \ldots + a_n \).

Step 3: Checking Boundedness

The inequality \( s_{n} \leqslant 1000 \) states that every partial sum \( s_n \) does not exceed 1000. This implies the sequence of partial sums is bounded above.

Step 4: Difference Between Terms and Partial Sums

Given that \( a_n \) are positive and \( s_{n} \leqslant 1000 \), each term adds a positive value to the previous sum but cannot surpass 1000, meaning the series does not diverge to infinity.

Step 5: Applying Convergence Test

According to the convergence criteria for series with positive terms, if the sequence of partial sums \( s_n \) is bounded above, then the series \( \Sigma a_{n} \) is convergent. The series approaches a finite limit as \( n \to \infty \).

Key Concepts

Partial Sums

In a series, partial sums play a crucial role in determining whether or not a series converges. A partial sum, denoted as \( s_n \), refers to the sum of the first \( n \) terms of the series. For example, for a series \( \Sigma a_{n} \), the \( n \)-th partial sum would be \( s_n = a_1 + a_2 + \ldots + a_n \). Partial sums help us visualize how the series develops as it incorporates more terms. Understanding the behavior of partial sums can give insight into the nature of the series, especially in relation to convergence or divergence. If the sequence of partial sums \( s_n \) approaches a specific finite number as \( n \) increases, then the series is convergent. Conversely, if the partial sums increase indefinitely as \( n \) grows, the series is divergent. This progression of sums thus provides a foundation for analyzing and interpreting the series behavior.

Bounded Series

A series is termed as bounded if there exists some constant value that the partial sums of the series do not exceed. Such a series implies that the sums are restrained within a particular range. In our given problem, the partial sums \( s_n \leq 1000 \) indicate that the series \( \Sigma a_{n} \) does not surpass the value of 1000, no matter how many terms we add. This bounded condition is significant because it is a critical indicator for convergence. If the sequence of partial sums of a series is bounded, and all terms are positive, it suggests that the series cannot grow indefinitely and therefore must approach a certain limit. This is closely tied to the convergence criteria that a bounded series with positive terms will converge to a finite limit.

Positive Terms

When talking about series, having positive terms implies that each term you add contributes to increasing the total sum. For example, in the series \( \Sigma a_{n} \), where all \( a_n > 0 \), any term added will result in a larger partial sum than the one before it. However, this does not necessarily mean the series diverges. The combination of positive terms and the bounded nature of the partial sums in our scenario is what's driving the convergence. Since the sums are increasing due to positive terms but also staying under a limiting boundary, it forms a 'bounded and increasing' sequence. By mathematical principle, a sequence that is both bounded and increasing must converge to a finite limit, supporting the fact that the series \( \Sigma a_{n} \) with positive terms and bounded partial sums will be convergent.