Math Is Fun Forum

AlexGo

Guest

Analysis Problems

I'm new to analysis and I'm having trouble with some (basic) problems.

Sketch the graphs of y = x and y = (x^4 + 1)/3, and thereby illustrate the behaviour of the real sequence a_n where a_n+1 = (a_n^4 + 1)/3. For which of the three starting cases a_1 = 0, a_1 = 1 and a_1 = 2 does the sequence converge? Now prove your assertion.

I've done the first part: clearly the sequence converges for a_1 = 0, and a_1 = 1, but diverges for a_1 = 2. I don't know where to start with a proof of this though? Any (small) hints would be greatly appreciated.

Thanks

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Analysis Problems

Hi AlexGo;

Don't remember a whole lot about spider web diagrams anymore but if you solve the equation.

x = (x^4 + 1) / 3 you will find that x = .337666765 and x= 1.307486100

For any a[1] > 1.307486100 the recurrence will diverge.

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