/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Let \(\left(x_{n}\right)\) be pr... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 6

Let \(\left(x_{n}\right)\) be properly divergent and let \(\left(y_{n}\right)\) be such that \(\lim \left(x_{n} y_{n}\right)\) belongs to \(\mathbb{R}\). Show that \(\left(y_{n}\right)\) converges to 0 .

Short Answer

Expert verified
Given the conditions stated in the question, the sequence \(y_n\) must converge to 0 for the product of sequences \(x_n y_n\) to converge to a real number even as \(x_n\) diverges properly.

Step by step solution

01

Recognize The Scenario

Recall that \(x_n\) is properly divergent, which means it tends towards infinity or negative infinity. Now, we know that the product of \(x_n\) and \(y_n\), denoted as \(x_n y_n\), is real and not infinite. This is a key piece of information.
02

Think through the Consequence

If \(y_n\) does not converge to 0, then as \(x_n\) tends towards infinity (or negative infinity), the product \(x_n y_n\) would also tend towards infinity (or negative infinity). However, we are given that \(x_n y_n\) converges to a real number, not infinity, which contradicts our assumption that \(y_n\) does not converge to 0.
03

Prove the Convergence of \(y_n\)

Given the conditions above, it must therefore be that \(y_n\) converges to 0 for the product \(x_n y_n\) to converge to a real number while \(x_n\) diverges properly.

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Most popular questions from this chapter

(a) Give an exarnple of a convergent sequence \(\left(x_{n}\right)\) of positive numbers with \(\lim \left(x_{n+1} / x_{n}\right)=1\). (b) Give an example of a divergent sequence with this property. (Thus, this property cannot be used as a test for convergence.)

Let \(x_{n}:=1 / \ln (n+1)\) for \(n \in \mathbb{N}\) (a) Use the definition of limit to show that \(\lim \left(x_{n}\right)=0\). (b) Find a specific value of \(K(\varepsilon)\) as required in the definition of limit for each of (i) \(\varepsilon=1 / 2\), and (ii) \(\varepsilon=1 / 10\).

Show that \(\lim \left(n^{2} / n !\right)=0\).

Let \(x_{1}:=a>0\) and \(x_{n+1}:=x_{n}+1 / x_{n}\) for \(n \in \mathbb{N}\). Determine if \(\left(x_{n}\right)\) converges or diverges.

Let \(\left(x_{n}\right)\) be a sequence of positive real numbers such that \(\lim \left(x_{n}^{1 / n}\right)=L<1 .\) Show that there exists a number \(r\) with \(0
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