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Part CCSIR NET June 2023recursive-sequence-fixed-point

Recursive sequence fixed point

Let be defined as . Given , define the sequence {} by and ₋ for n ≥ 1. Which of the following statements are true?

  1. A.If a = 0, then the sequence {} converges to 1/2.✓
  2. B.If a = 0, then the sequence {} converges to −1/2.
  3. C.The sequence {} converges for every a ∈ (−1/2, 3/2), and it converges to 1/2.✓
  4. D.If a = 0, then the sequence {} does not converge.

Solution

Write . Fixed points are ±1/2. For |a − 1/2| < 1 the distance satisfies ₊, which tends to 0, so for all a ∈ (−1/2, 3/2); a = 0 is included.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

Drill statements like this

Related counterexample: aₙ^{1/n} → L ⇒ aₙ₊₁/aₙ → L

More on this topic

From The Real Line › Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

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