Question

Check that the sequence {an} defined below is convergent and find its limit.

a₁ = 1, an+1 = 2 − 1/an, n≥1.

Answer 1

To check if the sequence {an} defined as a₁ = 1 and an+1 = 2 - 1/an, n≥1 is convergent, we will first show that the sequence is bounded. We can do this by proving that the sequence is always greater than or equal to 1. We will then use the Monotone Convergence Theorem to show that the sequence is convergent. The limit of the sequence is 1 because solving the recursive formula gives L² = 1.

Step-by-step explanation:

To check if the sequence {an} defined as a₁ = 1 and an+1 = 2 - 1/an, n≥1 is convergent, we will first show that the sequence is bounded. We can do this by proving that the sequence is always greater than or equal to 1. We will then use the Monotone Convergence Theorem to show that the sequence is convergent.

1. For n = 1, a₁ = 1. Since 1 ≥ 1, the base case is true.
2. We assume that for some positive integer k, ak ≥ 1. We will show that this implies ak+1 ≥ 1.
   
   ak+1 = 2 - 1/ak
   
   Since ak ≥ 1, 1/ak ≤ 1. Therefore, 2 - 1/ak ≥ 1. This means that ak+1 ≥ 1, which completes the induction step.
3. By the Monotone Convergence Theorem, a bounded increasing sequence is always convergent. Since the sequence is bounded above by 2 and bounded below by 1, the sequence {an} is both bounded and increasing. Therefore, it is convergent.

To find the limit of the sequence, we can take the limit of both sides of the recursive formula. Let L be the limit of the sequence. We have L = 2 - 1/L. Solving for L gives L² = 2 - 1, which simplifies to L² = 1. Taking the square root of both sides gives L = ±1. Since the sequence is always positive, the limit is L = 1.