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Chapter 9: Problem 31

Determine whether the statement is true or false. Explain your answer. If \(\sum u_{k}\) converges to \(L,\) then \(\sum\left(1 / u_{k}\right)\) converges to \(1 / L\)

Short Answer

Expert verified
The statement is false because \(\sum\left(1/u_k\right)\) does not converge.

Step by step solution

01

Understand the Given Series and Limits

The exercise gives us two series: \(\sum_{k=1}^{\infty} u_k\) which converges to \(L\), meaning that as more terms are added, the sum gets closer and closer to the value L. The statement asks whether the series \(\sum_{k=1}^{\infty} \left(1/u_k\right)\) also converges, but to \(1/L\).
02

Recall Properties of Convergent Series

If \(\sum u_k\) is convergent, it means that the terms \(u_k\) must approach zero as \(k\) approaches infinity. This is because for a series \(\sum a_k\) to converge, \(a_k\) must go to zero.
03

Investigate the Behavior of \(\sum\left(1/u_{k}\right)\)

The terms \(u_k\) approach zero, which implies that \(1/u_k\) grows without bound. For large \(k\), \(1/u_k\) will be very large, potentially making the series \(\sum\left(1/u_{k}\right)\) divergent, as each term will be getting larger and not approaching zero.
04

Conclude the Truth Value

Given that the terms \(\left(1/u_k\right)\) tend to increase as \(k\) increases, the series \(\sum\left(1/u_{k}\right)\) does not converge. Hence, it cannot converge to \(1/L\). Therefore, the statement "If \(\sum u_{k}\) converges to \(L,\) then \(\sum\left(1 / u_{k}\right)\) converges to \(1 / L\)" is false.

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Key Concepts

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

Divergence
Understanding the concept of divergence is essential when examining series. A series is said to diverge if the sum does not approach a finite limit as more terms are added. Instead, the series may increase indefinitely or behave erratically without settling towards a particular value.

In our exercise, we explore \[\sum \left( \frac{1}{u_{k}} \right)\]where the terms \(u_k\) of the initial series converge to zero. As \(u_k\) becomes small, \(\frac{1}{u_k}\) grows large. Thus, each term in the series \(\sum \left( \frac{1}{u_{k}} \right)\) becomes larger as we progress, which indicates that instead of converging, it diverges.

Key note:
  • Divergence means lack of settling towards a finite limit.
  • If \(a_k\) in series \(\sum a_k\) does not approach zero, the series diverges.
  • For \(\sum \left( \frac{1}{u_{k}} \right)\), growing terms suggest divergence.
Limit of a Sequence
The limit of a sequence is a fundamental idea to grasp when analyzing series. If the terms in a sequence \(a_k\) approach a specific number \(L\) as \(k\) becomes very large, we say that the sequence has a limit of \(L\).

This concept was used in determining the behavior of the series \(\sum u_k\). It specifies that as \(k\) increases, \(u_k\) should approach zero for the series \(\sum u_k\) to converge. If your sequence fails to have terms approaching zero, it will not contribute to a convergent series.

Keep in mind:
  • A sequence with a limit provides a predictable end behavior.
  • Terms of a convergent series shrink to zero.
  • Understanding limits help predict the behavior of sequences and series.
Properties of Series
Properties of series play a crucial role in determining whether a series converges or diverges. A key feature of a convergent series is that its terms approach zero; if this condition isn't met, the series cannot converge. This property directly informed the solution to our exercise.

When examining the series:\[\sum u_k \]we understand that convergence to a limit \(L\) implies diminishing term sizes \(u_k \to 0\). In contrast, the series:\[\sum \left(\frac{1}{u_k}\right)\]faces an opposite effect due to the inverse relationship. As \(u_k\) decreases, \(\frac{1}{u_k}\) grows, violating the requirement for convergence.

Key properties:
  • A convergent series' term approaches zero.
  • Inverse terms, like \(\frac{1}{u_k}\), will not satisfy convergence criteria if original terms are heading to zero.
  • Understanding how series and their terms behave is essential for determining their nature (convergent or divergent).

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Most popular questions from this chapter

Determine whether the series converges. $$ \sum_{k=1}^{\infty} \frac{1}{\sqrt{k^{2}+1}} $$

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