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Chapter 5: Problem 130

[T] Suppose that \(a_{n}\) is a sequence of positive numbers and the sequence \(S_{n}\) of partial sums of \(a_{n}\) is bounded above. Explain why \(\sum_{n=1}^{\infty} a_{n}\) converges. Does the conclusion remain true if we remove the hypothesis \(a_{n} \geq 0 ?\)

Short Answer

Expert verified
The series \(\sum_{n=1}^{\infty} a_n\) converges due to the bounded partial sums. If \(a_n\) includes negative terms, the conclusion may not hold.

Step by step solution

01

Understanding the Problem

We have a sequence of positive numbers \(a_n\) and a bounded sequence of partial sums \(S_n\). We need to show that \(\sum_{n=1}^{\infty} a_n\) converges by using these properties.
02

Analyzing Partial Sums

The partial sum sequence \(S_n = a_1 + a_2 + \cdots + a_n\) is given to be bounded above, which means there exists some number \(M\) such that \(S_n \leq M\) for all \(n\).
03

Applying Infinite Series Convergence Criterion

Since \(S_n\) is bounded above, the series \(\sum_{n=1}^{\infty} a_n\) must converge. For a series \(\sum a_n\) of non-negative terms, convergence follows if the sequence of partial sums is bounded.
04

Considering Negative Terms

If we remove the hypothesis \(a_n \geq 0\), the conclusion may not remain true. With negative terms, the sequence of partial sums may still be bounded without the series converging.

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Key Concepts

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

Bounded Sequence
A sequence is termed 'bounded' if its terms do not indefinitely increase or decrease beyond certain limits. In mathematical terms, a sequence \( \{x_n\} \) is bounded if there exists a real number \( M \) such that \(|x_n| \leq M \) for all values of \( n \). In this context, we consider the sequence of partial sums \( S_n \).
\[ S_n = a_1 + a_2 + \cdots + a_n \]
Since \( S_n \) is bounded above, it means there exists an upper limit \( M \) such that \( S_n \leq M \) no matter how many terms we add. This restriction implies that the series does not endlessly grow and therefore can have a finite limit, or, in other words, it converges.

  • The notion of boundedness provides a condition for a sequence approaching stability.
  • A bounded sequence helps ensure that an infinite series can be assessed for convergence given it doesn't surpass certain values.
Partial Sums
Partial sums play a fundamental role in understanding series. The concept of partial sums involves adding a finite number of initial terms of a sequence. For a series \( \sum_{n=1}^{fty} a_n \), the partial sums are given by \( S_n = a_1 + a_2 + \cdots + a_n \). Here, \( S_n \) represents sum totals up to the \( n^{th} \) term.

When analyzing the convergence of a series, looking at its partial sums can offer insights.
  • If the sequence of partial sums \( \{S_n\} \) converges to a limit, the infinite series itself converges.
  • Each partial sum is an approximation towards the infinite sum, and as more terms are added, it approaches a finite limit, provided the series is bounded.
In scenarios where \( S_n \) is shown to be bounded, like in the exercise, it indicates that our series has finite limit points, leading to the convergence of the series.
Non-negative Terms in Series
When dealing with series, especially in topics related to convergence, the presence of non-negative terms (\(a_n \geq 0 \)) introduces a special case. Non-negative terms in a series simplify the convergence analysis considerably.

Here's why non-negative terms are fair game:
  • Since each term \( a_n \geq 0 \), the sum \( S_n \) is non-decreasing, meaning each subsequent sum is equal to or larger than the previous sum.
  • For non-negative terms, if the sequence of partial sums \( \{S_n\} \) is bounded, it guarantees the series \( \sum_{n=1}^{fty} a_n \) converges.
However, without the non-negativity condition, the sequence might behave unexpectedly. When negative terms are present, a bounded sequence of partial sums does not necessarily imply convergence.
For instance, such a sequence could oscillate within a bounded range and yet not approach any particular value, thereby failing to converge.

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Most popular questions from this chapter

State whether each of the following series converges absolutely, conditionally, or not at all. $$\sum_{n=1}^{\infty}(-1)^{n+1}\left((n+1)^{2}-n^{2}\right)$$

Use the root test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$a_{k}=\frac{k^{e}}{e^{k}}$$

In the following exercises, use an appropriate test to determine whether the series converges. $$a_{k}=\left(\frac{k}{k+\ln k}\right)^{k}$$

In the following exercises, use an appropriate test to determine whether the series converges. $$a_{n}=\left(1+\frac{1}{n^{2}}\right)^{n}$$

In the following exercises, use an appropriate test to determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{(n-1)^{n}}{(n+1)^{n}}$$
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