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Chapter 9: Problem 57

Suppose that \(a_{n} > 0\) and \(b_{n} > 0\) for \(n \geq N(N\) an integer). If \(\lim _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)=\infty\) and \(\Sigma a_{n}\) converges, can anything be said about \(\Sigma b_{n} ?\) Give reasons for your answer.

Short Answer

Expert verified
The series \(\sum b_n\) diverges because \(b_n\) tends to zero slower than \(a_n\).

Step by step solution

01

Understand the Problem

We are given two sequences \(a_n\) and \(b_n\), both of which are positive for \(n \geq N\). The limit of the ratio \(\frac{a_n}{b_n}\) as \(n\) approaches infinity is infinity. Additionally, the series \(\sum a_n\) converges. We need to determine if anything can be concluded about the convergence or divergence of the series \(\sum b_n\).
02

Analyze the Limit Condition

The condition \(\lim_{n \to \infty} \frac{a_n}{b_n} = \infty\) implies that for sufficiently large \(n\), \(a_n\) is much larger than \(b_n\). This can be interpreted as \(b_n\) becoming negligibly small compared to \(a_n\) as \(n\) grows.
03

Recall Convergence Criteria for Series

A series \(\sum a_n\) converges if its terms \(a_n\) become small sufficiently quickly. In our case, \(\sum a_n\) is given to be convergent.
04

Connection Between Sequences

Since \(\sum a_n\) converges and \(\frac{a_n}{b_n}\) goes to infinity, \(b_n\) must tend to zero faster than \(a_n\) as \(n\) grows large. This rapid decrease in \(b_n\) alone doesn't ensure convergence.
05

Determine the Nature of \(\sum b_n\)

Since \(a_n\) becomes exceedingly larger than \(b_n\), the main factor influencing convergence or divergence of \(\sum b_n\) is whether \(b_n\) tends to zero fast enough. Because \(a_n\) diminishes rapidly enough for convergence, \(b_n\), being even smaller for large \(n\), implies that the series \(\sum b_n\) must diverge.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Series Divergence
A series is said to diverge if the sum of its terms does not approach a finite limit. This means that as we continue adding more terms, the total grows larger and does not settle at a fixed value. If you think of a series as a never-ending sum, a divergent series keeps increasing without bound.
Consider a series \( \sum b_n \). For it to diverge, there could be various reasons:
  • The terms \( b_n\) are not getting smaller fast enough, so when added together, they blow up.
  • Or, even if the terms tend to zero, they might do so too slowly.
In the given problem, since \( a_n \) dominates \( b_n \), and \( \sum a_n \) converges, while \( \sum b_n \) grows comparatively larger, \( b_n \) doesn鈥檛 shrink quickly enough, leading to \( \sum b_n \) diverging.
Comparison Test
The Comparison Test is a useful method to determine the convergence or divergence of series, using another series of known behavior. Here's how it works:
Given two series \( \sum a_n \) and \( \sum b_n \):
  • If \( 0 \leq a_n \leq b_n \) and \( \sum b_n \) converges, then \( \sum a_n \) also converges.
  • If \( 0 \leq b_n \leq a_n \) and \( \sum a_n \) diverges, then \( \sum b_n \) also diverges.
In the scenario provided, as \( n \) becomes large, \( a_n \) becomes significantly larger than \( b_n \), suggesting that \( b_n \) is dominated by \( a_n \). If \( a_n \) converges and diminishes fast enough, yet becomes exceedingly larger than \( b_n \), it hints towards the divergence of \( \sum b_n \). This inversion of typical comparison signals helps to deduce that when comparing terms, \( b_n \) must diverge.
Limit of a Sequence
Understanding the limit of a sequence is crucial to assessing behaviors of series over time. As \( n \) approaches infinity, if the limit of the sequence \( \left( \frac{a_n}{b_n} \right) = \infty \), it indicates that each \( a_n \) is becoming much larger than \( b_n \) for large \( n \).
This relationship between limits reveals important insights:
  • A sequence with terms unbounded, or exploding to infinity like in \( \frac{a_n}{b_n} \rightarrow \infty \), implies one term (here \( a_n \)) overshadows the other (\

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