Problem 24 - Calculus Single Variable

Question

Use either the word "may" or the word "must" to fill in the blank so that the completed sentence is correct. Explain your answer by referring to a theorem or example. If the partial sums of an infinite series are bounded, then the series __ converge.

Step-by-Step Solution

Answer

The series may converge.

Step 1: Understanding the Problem

We need to determine whether to use "may" or "must" in the statement "If the partial sums of an infinite series are bounded, then the series __ converge." The task is to appropriately complete the sentence based on a theorem or principle from calculus.

Step 2: Recall Relevant Theorem

Recall that a series \(\sum_{n=1}^{\infty} a_n\) is said to converge if the sequence of its partial sums \(S_n = a_1 + a_2 + a_3 + \ldots + a_n\) has a finite limit. This is the basic definition of convergence of a series.

Step 3: Examine Bounded Partial Sums

If the sequence of partial sums \(S_n\) is bounded, this means there exists some real number \(M\) such that \(|S_n| \leq M\) for all \(n\). However, boundedness does not necessarily imply that the limit exists. The series might oscillate or have some other behavior that prevents convergence.

Step 4: Reference Specific Example

Consider the alternating harmonic series \(\sum_{n=1}^{\infty} (-1)^{n+1}/n\), which is known to converge, and its partial sums are bounded. However, merely being bounded (like some divergent series) doesn't guarantee convergence without further conditions.

Step 5: Choose the Appropriate Word

Because bounded partial sums do not necessarily imply the limit exists, the correct completion of the sentence is: "If the partial sums of an infinite series are bounded, then the series **may** converge." Boundedness is a necessary but not sufficient condition for convergence.

Key Concepts

Partial Sums

Partial sums are an essential component when understanding the convergence of series. They represent the sum of the initial terms of a series up to a certain point. Consider a series denoted by \(\sum_{n=1}^{\infty} a_n\). The partial sums of this series, denoted as \(S_n\), are given by the sum of its first \(n\) terms: \( S_n = a_1 + a_2 + \cdots + a_n \).

Here's why partial sums matter: analyzing the behavior of partial sums as \(n\) approaches infinity helps determine whether the series converges to a certain value, or limit.

• If \(S_n\) approaches a specific finite number as \(n\) becomes very large, the series is called convergent.
• If not, the series diverges.

Understanding this concept is crucial for exploring more complex ideas around series convergence.

Bounded Sequences

A bounded sequence is one where all its terms lie within some specific interval on the real number line. This means there exists a real number \(M\) such that every term in the sequence is less than or equal to \(M\) in absolute value. Mathematically, this can be expressed as \(|S_n| \leq M\) for all \(n\).

When examining the convergence of a series, having bounded partial sums is a significant observation. However, it’s vital to remember that boundedness only indicates that partial sums do not grow indefinitely. It does not necessarily guarantee that these partial sums actually "settle down" or converge to a particular number.

This is why we say a series 'may' converge if its partial sums are bounded. Boundedness is a necessary condition, but on its own, it doesn’t confirm convergence without additional properties.

Alternating Harmonic Series

The alternating harmonic series is a classic example often discussed in the context of convergence. It is written as \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\). This series is known to converge, which means its partial sums tend to a particular limit as \(n\) becomes infinitely large.

The behavior of the alternating harmonic series is fascinating because, despite its terms decreasing slowly, the alternating positive and negative signs play a critical role in its convergence.

• Its partial sums are indeed bounded, which means they lie between two specific values on the number line.
• This series is an example where, even though the sums do not oscillate wildly or run off to infinity, the convergence is assured due to the alternating nature and the decreasing absolute value of its terms.

Studying such examples can deepen your understanding of how certain series behave and why specific conditions lead to convergence.

Convergent Series

A series is convergent if its partial sums approach a particular finite limit. This means that as you add more terms of the series, the total stabilizes at a certain number, rather than growing indefinitely or oscillating erratically.

To determine if an infinite series \(\sum_{n=1}^{\infty} a_n\) converges, consider if the sequence of partial sums \(S_n = a_1 + a_2 + ... + a_n\) has a finite limit.

• If the limit exists and is finite, the series is convergent.
• If not, the series diverges.

A useful series to compare is the geometric series, which helps understand convergence through its simple ratio test. Recognizing these patterns in series’ behaviors is important when applying fundamentals and theorems to various problems in calculus and beyond.