Suppose E an has partial = an 21. n=1,2,3,.. Sums Does the series converge ? Find an fn n7a. (Simplify !)
Suppose E an has partial = an 21. n=1,2,3,.. Sums Does the series converge ? Find an fn n7a. (Simplify !)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Topic: Infinite Series and Convergence**
**Problem Statement:**
Suppose \(\sum_{n=1}^{\infty} a_n\) has partial sums
\[ S_n = 2n, \, n = 1, 2, 3, \ldots \]
1. Does the series converge?
2. Find \(a_n\) for \(n \geq 2\). (Simplify!)
**Explanation:**
To determine if the series converges, we need to analyze the behavior of the partial sums, \(S_n\), as \(n\) approaches infinity. Given that \(S_n = 2n\), observe how this expression evolves for increasing values of \(n\).
To find the individual terms \(a_n\), recall the relationship:
\[ a_n = S_n - S_{n-1} \]
Apply this formula to derive \(a_n\) for \(n \geq 2\), ensuring that your computation simplifies the expression. Consider how this impacts the convergence or divergence of the series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85bb737b-6258-4d50-8ea7-e8c7d70d10bd%2Fffc84717-707d-4386-9034-fc65500ffc3f%2Fve4pdz8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Infinite Series and Convergence**
**Problem Statement:**
Suppose \(\sum_{n=1}^{\infty} a_n\) has partial sums
\[ S_n = 2n, \, n = 1, 2, 3, \ldots \]
1. Does the series converge?
2. Find \(a_n\) for \(n \geq 2\). (Simplify!)
**Explanation:**
To determine if the series converges, we need to analyze the behavior of the partial sums, \(S_n\), as \(n\) approaches infinity. Given that \(S_n = 2n\), observe how this expression evolves for increasing values of \(n\).
To find the individual terms \(a_n\), recall the relationship:
\[ a_n = S_n - S_{n-1} \]
Apply this formula to derive \(a_n\) for \(n \geq 2\), ensuring that your computation simplifies the expression. Consider how this impacts the convergence or divergence of the series.
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