Converging of a sequence

Question

Given a real sequence a_n, where a_n+1 = (a⁴_n + 1)/3. For which of the three starting cases a₁ = 0, a₁ = 1 and a₁ = 2 does the sequence converge? Now prove your assertion.

Answer 1

Do you mean a_n+1=(a_n⁴+1)/3?

Let's write out a few terms of the three series:

(1) a₁=0: 0, 1/3, 10/27, 82/243, ...

(2) a₁=1: 1, 2/3, 97/243, ...

(3) a₁=2: 2, 17/3, 83602/243, ...

It seems from observation that (3) definitely doesn't converge, while (1) and (2) could converge.

Consider (x⁴+1)/3<1, then x⁴+1<3, x⁴<2, x<1.19 (approx). So a₁ must be less than 1.19 for the series to converge, that is, for the nth term to converge to a finite value.

(1) and (2) satisfy this requirement, but (3) does not.