Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Why is this so for #3?
![**Question 3:**
If we know that
\[
\sum_{k=1}^{\infty} a_k = 10,000,
\]
then what can we say about
\[
\lim_{k \to \infty} a_k?
\]
---
**Explanation:**
This question is asking about a series and the behavior of its terms as \( k \) approaches infinity. Given that the infinite series of terms \( a_k \) sums to 10,000, we need to assess what happens to the individual terms \( a_k \) as \( k \) goes to infinity. This relates to the concepts of convergence and divergence in series analysis. If an infinite series converges, it is necessary for the limit of the terms \( a_k \) to be zero as \( k \to \infty \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61b75115-d70f-4fe2-af93-2076876ad69a%2F79178365-8bfe-468c-aa2a-a9ffd99ff934%2Fmymj2af_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 3:**
If we know that
\[
\sum_{k=1}^{\infty} a_k = 10,000,
\]
then what can we say about
\[
\lim_{k \to \infty} a_k?
\]
---
**Explanation:**
This question is asking about a series and the behavior of its terms as \( k \) approaches infinity. Given that the infinite series of terms \( a_k \) sums to 10,000, we need to assess what happens to the individual terms \( a_k \) as \( k \) goes to infinity. This relates to the concepts of convergence and divergence in series analysis. If an infinite series converges, it is necessary for the limit of the terms \( a_k \) to be zero as \( k \to \infty \).
Transcribed Image Text:### Section 10.4.3
We can be sure that \(\lim_{k \to \infty} a_k = 0\).
This statement implies that as \(k\) approaches infinity, the sequence \(a_k\) converges to zero. This concept is often used in the analysis of series to determine convergence properties, such as whether a series is convergent or divergent.
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