A rational map approaches two

u_1,u_2,u_3,\ldots

[2011/TPJC/Prelim/I/2]

Total marks: 6

Solution

(i)

Find the limit exactly

$u_1=1$ and $u_{n+1}=\frac{5u_n+4}{2u_n+3}$ . As $n\to\infty$ , $u_n\to l$ . Find $l$ exactly.

Explore this idea

Hint

Replace $u_n,u_{n+1}$ by $l$ and clear the denominator.
Reject the negative root.

Answer · (i)

l=2

Explore the idea

Climbing toward the fixed point

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 = 1

Iterations drawn4

Try an example

u(n = 5)

1.99938

u(n = 1)

Distance to l = 2

0.00063

Key idea

The curve is increasing and flatter than the diagonal at $2$ , so the terms creep monotonically to the fixed point.

Ideas to try

Add iterations. The web approaches

2

from one side without overshooting.

(ii)

Increment sign

By considering $y=\frac{5x+4}{2x+3}-x$ , show that $u_{n+1}<u_n$ if $u_n>l$ .

Explore this idea

Hint

Write $u_{n+1}-u_n$ as the height of the graph at $u_n$ .
Factorise the numerator and read the sign for $x>2$ .

Answer · (ii)

F(x)-x=-2(x-2)(x+1)/(2x+3)<0

Explore the idea

The increment is negative above 2

Switch to the increment plot: the dot's height is the next step. Above $x=2$ it is below the axis.

Starting valueu₁ = 4

Iterations drawn1

Try an example

u(n = 2)

2.18182

u(n = 1)

Distance to l = 2

0.18182

Key idea

A function that is negative above $2$ and positive below it pushes every term toward $2$ from either side.

Ideas to try

Open the increment plot and move the start across

2

. The dot crosses the axis exactly there.

(iii)

Behaviour from u₁ = 0

State the behaviour of the sequence when $u_1=0$ .

Explore this idea

Hint

Compute a few terms and compare with the fixed point.

Answer · (iii)

0

Explore the idea

From below

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

Iterations drawn5

Try an example

u(n = 6)

1.99964

u(n = 1)

Distance to l = 2

0.00036

Key idea

Whether the start is above or below $2$ , the same attracting fixed point decides the outcome.

Ideas to try

From

u_1=0

the web climbs the curve and levels off at

2

from below.

Takeaway

Sign of the increment

To show a sequence moves in one direction, show its one-step increment has a single sign on the whole region it can occupy.

A rational map approaches two

Solution

Find the limit exactly

Limit equation

Solve

Climbing toward the fixed point

Increment sign

Combine over a common denominator

Factorise

Read the sign for $u_n>2$

The increment is negative above 2

Behaviour from u₁ = 0

Terms

Conclusion

From below

Sign of the increment

Limit equation

Solve

Combine over a common denominator

Factorise

Read the sign for un>2u_n>2un​>2

Terms

Conclusion

Sign of the increment

Read the sign for $u_n>2$