Problem 86

Question

If $\sum a_{n}$ converges and $a_{n}>0$ for all $n,$ can anything be said about $\sum\left(1 / a_{n}\right) ?$ Give reasons for your answer.

Step-by-Step Solution

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Answer

$ \sum \left( \frac{1}{a_n} \right) $ diverges because $ a_n \to 0 $ implies $ \frac{1}{a_n} \to \infty.$

1Step 1: Understand the Given Information

We are given that the series $ \sum a_n $ converges and that each term $ a_n $ is positive. This means that the sequence $ a_n \rightarrow 0 $ as $ n \rightarrow \infty $.

2Step 2: Consider the Reciprocal Series

We need to evaluate the convergence of the series $ \sum \left( \frac{1}{a_n} \right) $. The task is to analyze whether this series converges, diverges, or whether no definite conclusion can be made.

3Step 3: Apply Limit Comparison Test Thoughts

If $ a_n $ approaches 0, then $ \frac{1}{a_n} $ becomes very large as $ n \rightarrow \infty $. Specifically, as $ a_n \to 0^+ $, $ \frac{1}{a_n} \to \infty $. This suggests that the terms $ \frac{1}{a_n} $ do not approach zero, indicating that the series may diverge.

4Step 4: Conclude Based on Harmonic Series Analogy

Since $ a_n \rightarrow 0 $ and $ \sum a_n $ converges, it implies that $ a_n $ converges to zero sufficiently fast. However, the harmonic series $ \sum \frac{1}{n} $ diverges even though $ \frac{1}{n} \to 0 $. Thus, $ \sum \left( \frac{1}{a_n} \right) $ diverges similarly due to terms becoming too large.

Key Concepts

Positive SequenceLimit Comparison TestHarmonic Series

Positive Sequence

In the realm of arithmetic sequences and series, understanding the nature of a positive sequence is crucial. A positive sequence is a series of terms, each of which is greater than zero.

When we say the series $ \sum a_n $ converges and each term $ a_n > 0 $, it implies a specific behavior. Each term getting progressively closer to zero as $ n \rightarrow \infty $. This is vital for convergence because, if the terms remained large, the series would not graduate to a finite sum.

The notion of positiveness ensures that each term adds positively to the sum.
Convergence suggests the terms decrease rapidly.
Positivity implies that there are no sign changes that affect summation.

With this in mind, when analyzing the reciprocal series like $ \sum \left( \frac{1}{a_n} \right) $, we need to consider how each positive $ a_n $ affects the behavior of its inverse.

Limit Comparison Test

The Limit Comparison Test is a mathematical tool used to determine the convergence or divergence of series. It is especially useful when dealing with reciprocal transformations, as seen in the problem.

This test compares the terms of a questionable series with those of a known reference, typically a simpler harmonic or geometric series. Here's how it works:

Assume two positive series $ \sum a_n $ and $ \sum b_n $.
Compute the limit $ \lim_{{n \to \infty}} \frac{a_n}{b_n} $.
If this limit is a positive finite number, then both series behave similarly—in the sense of convergence or divergence.

Applying this to $ \sum \left( \frac{1}{a_n} \right) $, where $ a_n \to 0 $, the terms $ \frac{1}{a_n} \to \infty $, resembling a divergent series. By comparing $ \frac{1}{a_n} $ terms to those of a harmonic series $ \sum \frac{1}{n} $, we find that they diverge due to the limiting behaviors.

Harmonic Series

The harmonic series plays an essential role as a standard reference in assessing convergence or divergence. It is the infinite series given by $ \sum \frac{1}{n} $, which remarkably diverges even as its terms approach zero.

In comparison with a series $ \sum \left( \frac{1}{a_n} \right) $, where $ a_n $ decreases to zero, each reciprocal term $ \frac{1}{a_n} $ becomes extremely large. Such large terms could mirror those of a harmonic series, signaling potential divergence.

The divergence of the harmonic series, despite its terms shrinking to zero, offers a potent analogy:

Like $ \sum \frac{1}{n} $, if $ \sum \left( \frac{1}{a_n} \right) $ contains unbounded terms, it hints at divergence.
Even when individual terms get smaller, their sum can still grow indefinitely.
This teaches that achieving convergence requires more than just decreasing term size—it demands a steep decrease at a rapid pace.

Understanding this principle can clarify why certain series fail to converge despite apparent decreases in individual term value.

Problem 85

Problem 87

Other exercises in this chapter

Problem 83

Which of the sequences $\left\\{a_{n}\right\\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left

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Problem 85

Show by example that $\sum\left(a_{n} / b_{n}\right)$ may converge to something other than $A / B$ even when $A=\Sigma a_{n}, B=\Sigma b_{n} \neq 0,$ and

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Problem 87

What happens if you add a finite number of terms to a divergent series or delete a finite number of terms from a divergent series? Give reasons for your answer.

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Problem 88

If $\sum a_{n}$ converges and $\sum b_{n}$ diverges, can anything be said about their term-by-term sum $\sum\left(a_{n}+b_{n}\right) ?$ Give reasons for y

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