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3. If we know that > ak = 10,000, then what can we way about lim ay? %3D k=1

3. If we know that > ak = 10,000, then what can we way about lim ay? %3D k=1

Calculus: Early Transcendentals
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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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 \( k \to \infty \).
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 \( k \to \infty \).
### 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.
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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