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Suppose that the series 1 ak converges absolutely. Prove that the series a converges.

Suppose that the series 1 ak converges absolutely. Prove that the series a converges.

Advanced Engineering Mathematics
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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**Problem Statement:** Suppose that the series \(\sum_{k=1}^{\infty} a_k\) converges absolutely. Prove that the series \(\sum_{k=1}^{\infty} a_k^2\) converges. --- **Explanation:** This problem is asking the student to show that if a series of numbers converges absolutely (i.e., the series remains convergent even when all its terms are replaced by their absolute values), then the series formed by squaring each term also converges. This is a common question in mathematical analysis related to series and sequence convergence, particularly focusing on the properties of absolute convergence. The problem requires a deeper understanding of series convergence and the manipulation of sequences. Concepts like the Cauchy-Schwarz inequality or comparison tests might be helpful to prove the statement.
Transcribed Image Text:**Problem Statement:** Suppose that the series \(\sum_{k=1}^{\infty} a_k\) converges absolutely. Prove that the series \(\sum_{k=1}^{\infty} a_k^2\) converges. --- **Explanation:** This problem is asking the student to show that if a series of numbers converges absolutely (i.e., the series remains convergent even when all its terms are replaced by their absolute values), then the series formed by squaring each term also converges. This is a common question in mathematical analysis related to series and sequence convergence, particularly focusing on the properties of absolute convergence. The problem requires a deeper understanding of series convergence and the manipulation of sequences. Concepts like the Cauchy-Schwarz inequality or comparison tests might be helpful to prove the statement.
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