Problem 26 - 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 diverge, then the series __ diverge.

Step-by-Step Solution

Answer

The series must diverge.

Step 1: Understand the Problem

We need to determine the correct word, "may" or "must," to fill in the sentence: 'If the partial sums of an infinite series diverge, then the series __ diverge.' The key term here is 'partial sums of an infinite series.'

Step 2: Recall the Definition of Series Convergence

Recall that an infinite series converges if its sequence of partial sums converges to a specific number. Conversely, if the sequence of partial sums does not converge, the series is said to diverge.

Step 3: Analyze the Given Condition

In this exercise, we are given that the partial sums of the infinite series diverge. This means that the sequence of partial sums does not approach any fixed value.

Step 4: Apply the Convergence-Divergence Relationship

According to the relationship between convergence and divergence, if the sequence of partial sums diverges, then the infinite series cannot converge. Thus, the series must diverge as well.

Step 5: Conclusion Based on Analysis

The correct word to fill in the blank is "must" because divergence of partial sums implies divergence of the series itself.

Key Concepts

Partial Sums

When we talk about an infinite series, it can be difficult to understand without breaking it down. A key element in understanding a series is the concept of partial sums. A partial sum is the sum of the first few terms of a series.
Think of it as taking smaller pieces of a puzzle to gradually see what the bigger picture might look like.

• The partial sum for the sequence \(a_1, a_2, a_3, \ldots\) is given by \(S_n = a_1 + a_2 + a_3 + \ldots + a_n\).
• As you increase the number of terms in the partial sum, \(S_n\), you're inching closer to understanding the behavior of the entire series.

Remember, whether an infinite series converges or diverges often hinges on the sequence of these partial sums.
When the partial sums approach a finite limit, we say the series converges.
If not, then the series diverges.

Series Convergence

The idea of series convergence is central when working with infinite series. A series converges if its sequence of partial sums approaches a specific, finite number as the number of terms grows indefinitely.

• If \(S_n\) tends to a number \(L\), i.e., \(\lim_{n \to \infty} S_n = L\), then the series is said to converge.
• This implies that as you keep adding more terms, the sum gets closer and closer to \(L\).

In essence, a convergent series has a total that can be pinned down to a particular value. This makes for a 'stable' outcome where the series can be summed up to a real number.
It's almost like watching a car that gradually comes to a stop as it approaches a red light, where finally the car (or the sum) rests on a definitive point.

Series Divergence

Series divergence occurs when the partial sums of a series do not settle down to a stable, finite value.
Instead, they may continue to increase or oscillate forever without approaching any particular number.

• If \(\lim_{n \to \infty} S_n\) does not exist or is infinite, the series diverges.
• This means the more terms you add, the larger or more chaotic the sum gets.

When partial sums diverge, it's like a car accelerating endlessly without ever stopping. It never comes to rest at any speed (or value) no matter how many miles it travels or how many terms you sum.