Show that the series Show that the series N=1 Cos N is divergent. E Σ (cos N)/N² is convergent. N=1
Show that the series Show that the series N=1 Cos N is divergent. E Σ (cos N)/N² is convergent. N=1
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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Question
Transcribed Image Text:**Problem Statement:**
1. Show that the series \(\sum_{N=1}^{\infty} \cos N\) is divergent.
2. Show that the series \(\sum_{N=1}^{\infty} \frac{\cos N}{N^2}\) is convergent.
**Explanation:**
- The first problem requires proving that the series of the cosine function evaluated at integer values is divergent.
- The second problem involves showing that the series formed by the cosine function divided by the square of the integers converges.
These exercises explore the concepts of convergence and divergence of infinite series, particularly involving trigonometric functions.
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Follow-up Question
Where does the subsequence series Nk = 2kpi come from?
While I know it's used to specify the set of solutions...how/why can we just pull it out and use it in this series?
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