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Chapter 2: Problem 5

Prove that if \(s_{n} \rightarrow \infty\) then \(\left(s_{n}\right)^{2} \rightarrow \infty\) also.

Short Answer

Expert verified
\((s_n)^2 \rightarrow \infty\) as \(s_n \rightarrow \infty\).

Step by step solution

01

Understanding the Problem

We need to prove that if a sequence \(s_n\) diverges to infinity, then the sequence \((s_n)^2\) also diverges to infinity. This means that for any arbitrarily large positive number \(M\), there exists a positive integer \(N\) such that for all \(n > N\), \(s_n > M\). Our goal is to show that for any arbitrarily large positive number \(M'\), there exists a \(N'\) such that for all \(n > N'\), \((s_n)^2 > M'\).
02

Setting Up the Logical Structure

Since \(s_n \to \infty\), for any large number \(M'\), we can find a point in the sequence after which every term is larger than \(M'^{1/2}\). So, given the same sequence \(s_n\), we choose \(M' = M\) and find a corresponding \(N\) such that for all \(n > N\), \(s_n > \sqrt{M'}\).
03

Proof by Direct Substitution

Given \(s_n > \sqrt{M'}\) for all \(n > N\), it follows that \((s_n)^2 > (\sqrt{M'})^2 = M'\). Thus, once \(s_n\) exceeds \(\sqrt{M'}\), \((s_n)^2\) will necessarily exceed \(M'\). This proves that \((s_n)^2 \to \infty\) as \(s_n \to \infty\).
04

Conclusion

The sequence \((s_n)^2\) diverges to infinity provided \(s_n\) diverges to infinity, concluding the proof as required.

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Key Concepts

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

Divergence to Infinity
When we talk about a sequence "diverging to infinity," we mean that its terms grow larger and larger without bound as we progress further along the sequence. In mathematical terms, for a sequence \(s_n\) to diverge to infinity, for any positive large number \(M\), there is a positive integer \(N\) such that for all \(n > N\), \(s_n > M\). This concept is crucial in understanding how sequences behave when they do not settle around a specific value.
  • It is important to grasp that divergence to infinity doesn't imply a specific speed of divergence鈥攋ust the direction and unbounded growth.
  • The sequence could diverge linearly, exponentially, or following any other pattern, as long as it eventually surpasses every finite boundary.
  • Recognizing divergence is key in predicting behavior of functions or sequences as they approach extremely large values.
In the given exercise, we explored how one sequence \(s_n\) diverging to infinity implies that the sequence \((s_n)^2\) also behaves in the same way, diverging to infinity.
Mathematical Proof
Mathematical proof is a logical method used to validate a statement or theorem beyond doubt. They involve logical reasoning and a series of arguments that connect known facts (axioms) to the conclusion.
  • Proofs can take various forms such as direct, contradiction, induction, and others tailored to the nature of the problem.
  • In our specific case, a direct proof was used, which means arguing directly from the assumption to the conclusion.
  • Starting with the assumption that \(s_n \to \infty\), we apply logical reasoning to show \((s_n)^2 \to \infty\).
Proofs are vital in mathematics to ensure that results are universally accepted and rely on an unbroken chain of reasoning. This chain often starts from well-established principles and moves step-by-step to new conclusions.
Real Analysis
Real Analysis is a branch of mathematics that deals with the behavior of real numbers, sequences, and functions. It forms the underpinning of many mathematical disciplines and provides a rigorous framework for understanding various concepts including convergence, limits, and continuity.
  • Real analysis looks at concepts such as limits of sequences and functions, and the convergence or divergence of these sequences and series.
  • It ensures that definitions are precise and results are derived from logical deductions following from these definitions.
  • In the topic above, we employed real analysis to formally define and reason about the divergence of a sequence and its squared version.
This branch of study is important because it validates results that we might intuitively believe to be true, grounding them in a firm foundation of logic and proof.

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Most popular questions from this chapter

If \(\left\\{s_{n}\right\\}\) is a sequence of positive numbers converging to 0 , show that \(\left\\{\sqrt{s_{n}}\right\\}\) also converges to zero.

Establish which of the following statements are true for an arbitrary sequence \(\left\\{s_{n}\right\\}\) (a) If all monotone subsequences of a sequence \(\left\\{s_{n}\right\\}\) are convergent, then \(\left\\{s_{n}\right\\}\) is bounded. (b) If all monotone subsequences of a sequence \(\left\\{s_{n}\right\\}\) are convergent, then \(\left\\{s_{n}\right\\}\) is convergent. (c) If all convergent subsequences of a sequence \(\left\\{s_{n}\right\\}\) converge to 0 , then \(\left\\{s_{n}\right\\}\) converges to 0 . (d) If all convergent subsequences of a sequence \(\left\\{s_{n}\right\\}\) converge to 0 and \(\left\\{s_{n}\right\\}\) is bounded, then \(\left\\{s_{n}\right\\}\) converges to \(0 .\)

Let \(\left\\{x_{n}\right\\}\) be a bounded sequence, let $$ y=\inf \left\\{x_{n}: n \in \mathbb{N}\right\\} \text { and } x=\sup \left\\{x_{n}: n \in \mathbb{N}\right\\} $$ Suppose that, moreover, \(y

Where possible find subsequences that are monotonic and subsequences that are convergent for the following sequences (a) \(\left\\{(-1)^{n} n\right\\}\) (b) \(\\{\sin (n \pi / 8)\\}\) (c) \(\\{n \sin (n \pi / 8)\\}\) (d) \(\left\\{\frac{n+1}{n} \sin (n \pi / 8)\right\\}\) (e) \(\left\\{1+(-1)^{n}\right\\}\) (f) \(\left\\{r_{n}\right\\}\) consists of all rational numbers in the interval \((0,1)\) arranged in some order.

Check the simple identity $$ \left(1+\frac{2}{n}\right)=\left(1+\frac{1}{n+1}\right)\left(1+\frac{1}{n}\right) $$ and use it to show that $$ \lim _{n \rightarrow \infty}\left(1+\frac{2}{n}\right)^{n}=e^{2} $$
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