Problem 61 - Thomas Calculus

Question

Suppose that \(a_{n}>0\) and $$\lim _{n \rightarrow \infty} a_{n}=\infty .$$ Prove that \(\sum a_{n}\) diverges.

Step-by-Step Solution

Answer

The series \(\sum a_n\) diverges because \(a_n\) grows without bound, making the partial sums grow indefinitely.

Step 1: Recall the Definitions

First, let's recall the definitions: A sequence \(a_n\) is said to diverge to infinity if for every positive number \(M\), there exists an index \(N\) such that \(a_n > M\) for all \(n > N\). A series \(\sum a_n\) diverges if the sequence of partial sums diverges.

Step 2: Understand the Given Limit

We are given that \(\lim _{n \rightarrow \infty} a_{n} = \infty\). This means that for every positive number \(M\), there exists a number \(N\) such that \(a_n > M\) for all \(n > N\). This implies that the terms of the sequence \(a_n\) can grow without bound.

Step 3: Analyze Partial Sums of \( \,\sum a_n \)

Consider the partial sums \(S_k = a_1 + a_2 + ... + a_k\). Since \(a_n\) becomes arbitrarily large as \(n\) increases, each subsequent term \(a_n\) in the series adds a large positive value to the partial sum \(S_k\).

Step 4: Prove Divergence of the Series

For a series \(\sum a_n\) to converge, the sequence of partial sums \(S_k\) must approach a finite limit as \(k\) tends to infinity. Here, since \(a_n\) grows without bound (i.e., for any predetermined large number, \(M\), \(a_n\) can be made larger than \(M\)), the sequence of partial sums \(S_k\) will also grow without limit as we keep adding increasingly larger terms. Hence, \(\sum a_n\) diverges.

Key Concepts

Partial Sums

Partial sums are an essential concept when analyzing the behavior of a series. They represent the sum of the first few terms in a series. For a sequence of numbers given by \( a_1, a_2, a_3, \ldots \), the partial sum \( S_k \) is the sum of the first \( k \) terms: \( S_k = a_1 + a_2 + \cdots + a_k \).

Think of partial sums as a way to "build up" the series term by term. Just by adding a few first terms together—and then a few more—you gradually create a clearer picture of the series' behavior.

• If the sequence of partial sums \( S_k \) approaches a specific number as \( k \) gets larger and larger, the series is said to converge.
• However, if the partial sums diverge, meaning \( S_k \) grows without bounds, the series itself diverges.

Understanding partial sums is key to grasping whether a series overall settles down to a limit or not.

Infinite Sequences

Infinite sequences are a progression of numbers ordered in an endless manner. Each number in the sequence is referred to as a term, and sequences follow a specific rule or pattern. In our original exercise, the sequence is \( a_1, a_2, a_3, \ldots \), where each \( a_n \) is greater than zero and grows without bound as \( n \) increases.

An infinite sequence may:

• Converge to a limit, which means as you advance through the terms of the sequence, they approach a particular value.
• Diverge, where the terms do not settle to a particular number as they extend indefinitely.

In the context of our exercise, we determined that because each \( a_n \) grows beyond any fixed positive number, the sequence diverges. The crucial part is acknowledging that this unbounded growth has significant implications for behavior when these terms are summed.

Series Convergence and Divergence

The core concept distinguishing convergent and divergent series lies in the behavior of the sum of an infinite sequence’s terms. A series is an accumulation of terms from a sequence, and for the series \( \sum a_n \) to converge, the corresponding sequence of partial sums \( S_k \) must approach a finite number as \( k \) goes to infinity.

A convergent series is like a balancing act where, as more terms accumulate, they settle calmly around a specific value. In contrast, a divergent series reflects growing, unbounded behavior where the accumulated terms never zero into one number but instead march on infinitely.

• Convergence occurs when the organizing rule or pattern of the sequence ensures the terms diminish enough to keep the sum bounded.
• Divergence means there's no stop in sight—terms can be large, and their accumulation grows without a limit.

In our case, given that \( \lim _{n \to \infty} a_n = \infty \), we see that each term is too large for the series to ever settle into a limit, leading directly to divergence. This understanding of convergence versus divergence is pivotal in assessing series behavior in calculus.