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Let a 20 be a real number (to be chosen below) and recursively define an-1 + a2-1 an = 3 for n > 2. C. On the other hand, let a₁ = 3. Show that the sequence [an]nen diverges. HINT. First prove a lower bound of the form an Z f(n) for some function f: IN → R you understand better.

Let a 20 be a real number (to be chosen below) and recursively define an-1 + a2-1 an = 3 for n > 2. C. On the other hand, let a₁ = 3. Show that the sequence [an]nen diverges. HINT. First prove a lower bound of the form an Z f(n) for some function f: IN → R you understand better.

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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Prove the following step by step please, with focus on proof writing style especially

Let a₁0 be a real number (to be chosen below) and recursively define an-1 + an-1 3 for n > 2. C. d. an = On the other hand, let a₁ = 3. Show that the sequence {an}neN diverges. HINT. First prove a lower bound of the form an ≥ f(n) for some function ƒ: N → R you understand better. Find all a₁ ≥ 0 for which the sequence {an}neN converges and deter- mine the limit in each case. HINT. First find some a₁ ≥ 0 for which {an}neN converges to a positive real number.
Transcribed Image Text:Let a₁0 be a real number (to be chosen below) and recursively define an-1 + an-1 3 for n > 2. C. d. an = On the other hand, let a₁ = 3. Show that the sequence {an}neN diverges. HINT. First prove a lower bound of the form an ≥ f(n) for some function ƒ: N → R you understand better. Find all a₁ ≥ 0 for which the sequence {an}neN converges and deter- mine the limit in each case. HINT. First find some a₁ ≥ 0 for which {an}neN converges to a positive real number.
Expert Solution
Step 1: Writing down the given information

Given that a1≥0 and recursively defined an=an−1+an−133 for n≥2.

Note that the given sequence is positive and monotonically increasing.

(c)

Let a1=3. We need to show that the sequence {an}n∈N diverges.

For this, we will find a function f(n) such that an≥f(n) and f:N→R.

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