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Prove that 1 1 1/1 + 2√2+3+√3+...+ 1 <3- n√n 2 √n for all n ≥ 1. Prove that the sequence on the left side of the equation converges. Let K> 1 be a real number. Consider the sequence £₁ = 1 and 2n+1 = 2KK for all n ≥ 1. Prove that for all n we have r² ≤ K and în ≤ En+1. Prove that (rn) converges to VK.

Prove that 1 1 1/1 + 2√2+3+√3+...+ 1 <3- n√n 2 √n for all n ≥ 1. Prove that the sequence on the left side of the equation converges. Let K> 1 be a real number. Consider the sequence £₁ = 1 and 2n+1 = 2KK for all n ≥ 1. Prove that for all n we have r² ≤ K and în ≤ En+1. Prove that (rn) converges to VK.

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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(d) Prove that #11 1 1V₁ + 2√2 + 3√3 53-√₁ <3- + 1 + n√n for all n ≥ 1. Prove that the sequence on the left side of the equation converges. (e) Let K> 1 be a real number. Consider the sequence ₁ = 1 and In+1 Prove that for all n we have r² ≤ K and En ≤n+1. Prove that (n) 2K for all n ≥ 1. 2²+K converges to VK.
Transcribed Image Text:(d) Prove that #11 1 1V₁ + 2√2 + 3√3 53-√₁ <3- + 1 + n√n for all n ≥ 1. Prove that the sequence on the left side of the equation converges. (e) Let K> 1 be a real number. Consider the sequence ₁ = 1 and In+1 Prove that for all n we have r² ≤ K and En ≤n+1. Prove that (n) 2K for all n ≥ 1. 2²+K converges to VK.
Expert Solution
Step 1

In this question, we need to show that the given sequences are convergent.

We know that a monotone bounded sequence is a convergent sequence and converges to its supremum.

For part (d), the given sequence is an=111+122+133+…+1nn.

For part (e), the given sequence is x1=1 and xn+1=2Kxnxn2+K for all n>1, where K>1.

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