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Chapter 8: Problem 176

(The \(n\) th Term Test) Show that if \(\sum_{n=1}^{\infty} a_{n}\) converges then \(\lim _{n \rightarrow \infty} a_{n}=0\).

Short Answer

Expert verified
The given statement is true. If a series converges, then the limit of its sequence of terms must be zero, which is what \(n\) th Term Test states. This is proven by assuming the contrary and showing it leads to a contradiction.

Step by step solution

01

Understanding the Problem

The problem is asking to prove that if a series converges, then the limit of its sequence of terms must be equal to zero. In mathematical notation, if \(\sum_{n=1}^{\infty} a_{n}\) converges, then \(\lim _{n \rightarrow \infty} a_{n}=0\). We will assume the contrary to prove it.
02

Assume the Contrary

We will assume the contrary, which is \(\lim _{n \rightarrow \infty} a_{n}\) does not equal zero and that there exists a positive number \( \varepsilon \) such that \( \left| a_{n} \right| \geq \varepsilon > 0\) for an infinite number of \(n\)'s.
03

Apply Assumption to The Series

Apply the assumption to the series \(\sum_{n=1}^{\infty} a_{n}\), it is implied that we have an infinite number of terms in our series which are greater than or equal to \( \varepsilon > 0 \) in an absolute sense. This means our series \(\sum_{n=1}^{\infty} a_{n}\) is greater or equal to the sum of an infinite number of \( \varepsilon \) which equates to infinity. Therefore, \(\sum_{n=1}^{\infty} a_{n} \geq \infty\), which means the series diverges.
04

Contradiction to Our Initial Assumption

This result contradicts our initial assumption that the series \(\sum_{n=1}^{\infty} a_{n}\) converges. Therefore, the assumption that \(\lim _{n \rightarrow \infty} a_{n}\) does not equal zero is incorrect.
05

Conclusion

Therefore, if the series \(\sum_{n=1}^{\infty} a_{n}\) converges, then the limit of its sequence of terms must be equal to zero, \(\lim _{n \rightarrow \infty} a_{n}=0\)

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Key Concepts

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

Real Analysis
Real Analysis is a branch of mathematics dealing with the behavior and properties of real numbers, sequences, and series. It鈥檚 foundational to understanding various mathematical concepts and proving theorems that are crucial in analyzing the convergence and divergence of series.

One important aspect of Real Analysis is exploring the rules and conditions under which mathematical series converge to a certain value. The exercise given is a common question in Real Analysis, prompting students to demonstrate an understanding of the nth Term Test for Convergence. Here, the exercise tests the premise that for a series to converge, its sequence's terms must approach 0 as the sequence proceeds to infinity.

When carrying out a Real Analysis, it's critical to maintain logical clarity and adhere to strict proof structures, as seen in the step-by-step solution provided. Moreover, failing to understand these core principles can lead to misconceptions and errors in proofs, making the mastery of these concepts critical for students.
Series Convergence
Series convergence is the idea that as more and more terms are added to a sequence, the sum approaches a fixed value. This is a concept every student of calculus and analysis will encounter. The nth Term Test, as highlighted in the exercise, is a tool to determine if a series converges or not.

In the nth Term Test, we examine the limit of the sequence of terms in a series. If this limit does not approach zero, the series does not converge. This concept is pivotal because if individual terms do not become infinitesimally small, their sum cannot stabilize to a finite value. This is illustrated in the provided problem, which efficiently ties the limit of a sequence to the broader idea of series convergence. The convergence of series is extensively used in various areas of mathematics and science, especially in analyzing the behavior of functions and solving differential equations.
Limit of a Sequence
The limit of a sequence is a fundamental concept in mathematical analysis that deals with the behavior of sequences as they progress towards infinity. The term 'limit' refers to the value that the terms of the sequence tend to approach as the index (usually denoted as 'n') becomes very large.

Understanding the limit of a sequence is essential for grasping deeper topics in calculus and Real Analysis, including series convergence. According to the problem, the limit of a sequence \(a_n\) as n approaches infinity, denoted \(\lim _{n \rightarrow \infty} a_{n}\), must be 0 for the series to be convergent. Hence, this exercise teaches a critical aspect of limits: if the terms of a sequence do not get infinitely close to zero, the series constituted by these terms cannot sum up to a finite value. The limit concept is not only essential for ensures proper understanding of convergence but also plays a significant role in defining continuity, derivatives, and integrals in calculus.

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Most popular questions from this chapter

Problem 190. Suppose the power series \(\sum a_{n} x^{n}\) has radius of convergence \(r\) and the series \(\sum a_{n} r^{n}\) converges absolutely. Then \(\sum a_{n} x^{n}\) converges uniformly on \([-r, r] .\) [Hint: For \(\left.|x| \leq r,\left|a_{n} x^{n}\right| \leq\left|a_{n} r^{n}\right| .\right]\)

(The Strong Cauchy Criterion) Show that \(\sum_{k=1}^{\infty} a_{k}\) converges if and only if \(\lim _{n \rightarrow \infty} \sum_{k=n+1}^{\infty} a_{k}=0 .\)

Observe that for all \(x \in[-1,1]|x| \leq 1 .\) Identify which of the following series converges pointwise and which converges uniformly on the interval \([-1,1] .\) In every case identify the limit function. (a) \(\sum_{n=1}^{\infty}\left(x^{n}-x^{n-1}\right)\) (b) \(\sum_{n=1}^{\infty} \frac{\left(x^{n}-x^{n-1}\right)}{n}\) (c) \(\sum_{n=1}^{\infty} \frac{\left(x^{n}-x^{n-1}\right)}{n^{2}}\) Using the Weierstrass- \(M\) test, we can prove the following result.

Prove the Cauchy criterion. At this point several of the tests for convergence that you probably learned in calculus are easily proved. For example:

Prove Theorem 35. [Hint: \(\left|s_{m}-s_{n}\right|=\left|s_{m}-s+s-s_{n}\right| \leq\) \(\left.\left|s_{m}-s\right|+\left|s-s_{n}\right| \cdot\right]\) So any convergent sequence is automatically Cauchy. For the real number system, the converse is also true and, in fact, is equivalent to any of our completeness axioms: the NIP, the Bolzano-Weierstrass Theorem, or the LUB Property. Thus, this could have been taken as our completeness axiom and we could have used it to prove the others. One of the most convenient ways to prove this converse is to use the Bolzano-Weierstrass Theorem. To do that, we must first show that a Cauchy sequence must be bounded. This result is reminiscent of the fact that a convergent sequence is bounded (Lemma 2 of Chapter 4\()\) and the proof is very similar.
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