A conditional limit in a two-cycle map

A sequence of positive real numbers satisfies

xn+1=3−xn2xn+1,n≥1.x_{n+1}=\frac{3-x_n}{2x_n+1},\quad n\ge1.

(i) Given convergence to ll, find ll exactly. marks

(ii) Using a sketch of y=3−x2x+1−xy=\frac{3-x}{2x+1}-x and considering xn+1−xnx_{n+1}-x_n, prove xn+1>xnx_{n+1}>x_n if 0<xn<l0<x_n<l, and xn+1<xnx_{n+1}<x_n if xn>lx_n>l. marks

[2011/CJC/Prelim/I/2]

Total marks: 6

Solution

(i)

Exact limit

Positive xnx_n satisfy xn+1=3−xn2xn+1x_{n+1}=\frac{3-x_n}{2x_n+1}. Given that the sequence converges to ll, find ll exactly.

Explore this idea

Hint

  1. Replace both terms by ll and clear the denominator.

  2. Use the quadratic formula and choose the positive root.

Answer · (i)

l=(7−1)/2l=(\sqrt7-1)/2.

The convergence premise forces the constant sequence: F∘F is the identity, so nonconstant admissible sequences form two-cycles.

Explore the idea

The curve meets the diagonal once

Start at the chosen value on the horizontal axis, go up to the curve, across to the diagonal, and repeat. Each corner of the web is the next term.

Starting valueu₁ = 0.3
Iterations drawn4
Try an example
u(n = 5)
0.3
u(n = 1)
0.3
Distance to l ≈ 0.823
0.52288
Key idea

A fixed point exists and is unique, but the web around it does not spiral inward. It closes into a rectangle.

Ideas to try
  1. Spot the single crossing of the curve and the diagonal: that is ll.
  2. Start away from it and add iterations. What shape does the web make?
(ii)

Increment sign from a sketch

Using a sketch of y=3−x2x+1−xy=\frac{3-x}{2x+1}-x and considering xn+1−xnx_{n+1}-x_n, prove xn+1>xnx_{n+1}>x_n if 0<xn<l0<x_n<l, and xn+1<xnx_{n+1}<x_n if xn>lx_n>l.

Explore this idea

Hint

  1. Identify the graph's height with xn+1−xnx_{n+1}-x_n.

  2. Read its sign from the sketch: positive before ll, negative after.

y = (3−x)/(2x+1) − x, x > 0
y = (3−x)/(2x+1) − x, x > 0Scroll across to see the full diagram.

Answer · (ii)

F(x)−x=(3−2x−2x2)/(2x+1)F(x)-x=(3-2x-2x^2)/(2x+1): positive on (0,l)(0,l), zero at ll, negative for x>lx>l.

This one-step comparison does not imply convergence.

Explore the idea

One step moves toward l, then overshoots

Open the increment plot. The sign of the dot's height is the direction of the first step.

Starting valueu₁ = 0.3
Iterations drawn1
Try an example
u(n = 2)
1.6875
u(n = 1)
0.3
Distance to l ≈ 0.823
0.86462
Key idea

Each single step is correct, but each step also overshoots ll by exactly as much as it undershot, so the web only ever visits two values.

Ideas to try
  1. With one iteration from 0.30.3 the term jumps from 0.30.3 to 1.68751.6875: an increase, as the sketch says.
  2. Add a second iteration. It lands back on 0.30.3: the web has closed.
Takeaway

A true step is not a trend

The two printed inequalities describe each step correctly. Because F(F(x))=xF(F(x))=x, a nonconstant admissible sequence alternates between two values, so the premise of convergence in (i) forces xn=lx_n=l for every nn.