Define (xn)n≥1 by x1 := 0 and xn+1 := (xn + 3)/4 for n ≥ 1. Prove that the sequence (xn)n≥1 isconvergent and calculate the limit limn→∞ xn

Question

Define (xn)n≥1 by x1 := 0 and xn+1 := (xn + 3)/4 for n ≥ 1. Prove that the sequence (xn)n≥1 isconvergent and calculate the limit limn→∞ xn
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Solution 1

To prove that the sequence (xn)n≥1 is convergent, we can use the Monotone Convergence Theorem. This theorem states that every bounded and monotone sequence is convergent.

Step 1: Prove that the sequence is increasing (monotone)

We can prove this by induction. For n=1, x1=0 and x2=(x1+3)/4=3/4. So, Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study prob

Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
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