/*! 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 1 Let \(\sum a_{n}\) be a given se... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 1

Let \(\sum a_{n}\) be a given series and let \(\sum b_{n}\) be the series in which the terms are the same and in the same order as in \(\sum a_{n}\) except that the terms for which \(a_{n}=0\) have been omitted. Show that \(\sum a_{n}\) converges to \(A\) if and only if \(\sum b_{n}\) converges to \(A\).

Short Answer

Expert verified
A series \(\sum a_{n}\) and \(\sum b_{n}\), where \(\sum b_{n}\) omits the terms where \(a_{n}=0\), will both converge to the same value (in this case, \(A\)) if any one of them does. The zero terms in \(\sum a_{n}\) have no material impact on the limit.

Step by step solution

01

Define the Series

Given two series, \(\sum a_{n}\) and \(\sum b_{n}\), where series \(\sum b_{n}\) is same as \(\sum a_{n}\) but the terms for which \(a_{n}=0\) have been omitted. We need to prove that both the series converge to the same number A.
02

Assume Convergence

Let's start by assuming that \(\sum a_{n}\) converges to A. This implies that for every number \(蔚 > 0\), there is an integer \(N1\) such that \(| a_{n} - A | < 蔚 \) for all \( n > N1\).
03

Prove \(\sum b_{n}\) Converges

Since all the non-zero terms of series \(\sum a_{n}\) are in series \(\sum b_{n}\), the convergence of \(\sum a_{n}\) to A implicates the same convergence for \(\sum b_{n}\), because \( a_{n} = b_{n} \) amiss those terms for which \( a_{n} = 0\). Hence, \(\sum b_{n}\) converges to A.
04

Assume \(\sum b_{n}\) Converges

On the other hand, assume that \(\sum b_{n}\) converges to A. In this case, series \(\sum a_{n}\) will also converge to A, because the zero terms have no impact on the limit.

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Most popular questions from this chapter

If \(0

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

Show directly from the definition that the following are not Cauchy sequences. (a) \(\left((-1)^{n}\right)\), (b) \(\left(n+\frac{(-1)^{n}}{n}\right)\), (c) \((\ln n)\).

Show that if \(\left(x_{n}\right)\) is unbounded, then there exists a subsequence \(\left(x_{n_{k}}\right)\) such that \(\lim \left(1 / x_{n_{2}}\right)=0\).

Establish the convergence or the divergence of the sequence \(\left(y_{n}\right)\), where $$y_{n}:=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n} \quad \text { for } \quad n \in \mathbb{N}$$
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