Chapter 8.1, Problem 45E

a.

To determine

To Find : If the sequence is convergent or divergent.

Answer to Problem 45E

The sequence obtained is a divergent sequence.

Explanation of Solution

Given Information :

  $a_1=1,a_{n+1}=4-a_n\text{for}n\ge 1$.

As, $a_1=1,a_{n+1}=4-a_n\text{for}n\ge 1$

for $a_1=1,a_2=4-a_1=3$

  $a_3=4-a_2=1$

  $a_4=4-a_3=3$

Therefore, the sequence is : 1, 3, 1, 3, 1, 3, 1, 3 . . . . .

The elements of the sequence are oscillating between 1 and 3 without approaching any number also limit does not exist for this sequence.

Thus, The obtained sequence is a divergent sequence.

b.

To determine

To Find : If the sequence is convergent or divergent when, $a_1=2$ .

Answer to Problem 45E

The obtained sequence is convergent sequence.

Explanation of Solution

Given Information :

  $a_1=2,a_{n+1}=4-a_n\text{for}n\ge 1$

As $a_1=2,a_{n+1}=4-a_n\text{for}n\ge 1$

For $a_1=2,a_2=4-2=2$

  $a_2=2,a_3=4-2=2$

Thus the sequence obtained is : 2, 2, 2, 2, . . . . .

Therefore,

The obtained sequence is Convergent with a limit of 2.