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8. Provide justification for you what will appear on the ex asked. (-1)" (2n)! (a) n=1 n4 – 4n -

8. Provide justification for you what will appear on the ex asked. (-1)" (2n)! (a) n=1 n4 – 4n -

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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How would you prove 7(a)?

Here's the transcription and explanation for the content related to an educational setting: --- ### Mathematical Series and Convergence **6. Series Analysis** (a) Prove that \(\sum_{k=0}^{\infty} ar^k = \frac{a}{1 - r}\) if and only if \(|r| < 1\). (b) Prove that \(\sum_{k=0}^{\infty} ar^k\) diverges whenever \(|r| \geq 1\). **7. Series Convergence** Determine whether the following series converge absolutely, conditionally, or diverge. Provide justification for your conclusions. These examples might not be exactly what will appear on your exam, but they give you an idea of typical questions. (a) \(\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\) (b) \(\sum_{n=8}^{\infty} \frac{n^4 - 4n}{n^6 + 9n^2 + 3}\) (c) \(\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\) (d) \(\sum_{n=1}^{\infty} \left(\frac{-1}{2}\right)^n\) **8. Proving Convergence** Prove that if \(\sum_{k=1}^{\infty} a_k\) converges, then \(a_k \to 0\). **9. Subset Lengths** Let \(\ell(A)\) denote the length of \(A \subseteq \mathbb{R}\). (a) Prove that \(A \subseteq B\) implies \(\ell(A) \leq \ell(B)\). --- This content focuses on topics related to mathematical series, showcasing how to determine convergence or divergence, and highlights fundamental theorems related to series and set lengths.
Transcribed Image Text:Here's the transcription and explanation for the content related to an educational setting: --- ### Mathematical Series and Convergence **6. Series Analysis** (a) Prove that \(\sum_{k=0}^{\infty} ar^k = \frac{a}{1 - r}\) if and only if \(|r| < 1\). (b) Prove that \(\sum_{k=0}^{\infty} ar^k\) diverges whenever \(|r| \geq 1\). **7. Series Convergence** Determine whether the following series converge absolutely, conditionally, or diverge. Provide justification for your conclusions. These examples might not be exactly what will appear on your exam, but they give you an idea of typical questions. (a) \(\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\) (b) \(\sum_{n=8}^{\infty} \frac{n^4 - 4n}{n^6 + 9n^2 + 3}\) (c) \(\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\) (d) \(\sum_{n=1}^{\infty} \left(\frac{-1}{2}\right)^n\) **8. Proving Convergence** Prove that if \(\sum_{k=1}^{\infty} a_k\) converges, then \(a_k \to 0\). **9. Subset Lengths** Let \(\ell(A)\) denote the length of \(A \subseteq \mathbb{R}\). (a) Prove that \(A \subseteq B\) implies \(\ell(A) \leq \ell(B)\). --- This content focuses on topics related to mathematical series, showcasing how to determine convergence or divergence, and highlights fundamental theorems related to series and set lengths.
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