**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

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2. Let an be the sequence defined inductively by a₁ = 2 and an+1 (a) Prove by induction that an € [1,2] for all n € N. (b) Prove that a² ≥ 2 for all n € N. = 1/2 (a₁ +²2²) an An (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (by (a)) so it follows, by the mono- tone convergence theorem, that (an) converges to some limit L. Show that L > 0 and L² = 2.
Prove the statement
a < b be real numbers and assume a sequence (Xk)1 is strictly decreasing and Xk E (a, b) for every natural number k. Prove that the sequence (xk)1 is convergent. Let
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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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1 2. Let an be the sequence defined inductively by a₁ = 2 and an+1 = (a) Prove by induction that an € [1, 2] for all n € N. (b) Prove that a ≥2 for all ne N. an + an (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (by (a)) so it follows, by the mono- tone convergence theorem, that (an) converges to some limit L. Show that L>0 and L² = 2.

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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.