**3. Prove the root test**: Let \(\{a_n\}\) be a sequence of real numbers, and define \[\rho(n) = |a_n|^{1/n}.\]Prove the following:(i) If there exists \(c < 1\) and \(N \geq 1\) so that \(\rho(n) \leq c\) for all \(n \geq N\), then \(\sum a_n\) converges absolutely.(ii) If for all \(N\) there exists \(n \geq N\) so that \(\rho(n) \geq 1\), then \(\sum a_n\) diverges.

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Answered: 3. Prove the root test: Let {an} be a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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2. Suppose that the sequence {an} converges to a and that a 0. Then use the Binomial Formula to show that || ≤nd ≤ for every index n
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3. Prove the root test: Let {an} be a sequence of real numbers, and define p(n) = |a,|'/n. Prove the following: (i) If there exists c <1 and N > 1 so that p(n) < c for all n > N, then Ean converges absolutely. (ii) If for all N there exists n > N so that p(n) > 1, then an diverges.