I received a Whatsapp message from student. He asked me whether the following shortcut is useful to determine whether the iteration is converges.

After research, the conclusion is the formula is useful only if you know the real root (the final answer).

## According to Pelangi book (wrong)

If is an **approximation** to a root of the equation , then sequence defined by with a starting value close to will converge to as long as .

## The Real

If is **a real root** of the equation , then sequence defined by with a starting value close to will converge to as long as .

I am not going to describe the proving in details. For more details, check here.

### Sample Question from school mock exam

### Solution given by the teacher

### Solution b

Seriously, i wont suggest students to write this working in STPM exam. Even the F'(x) is more than 1, where x is the first estimate, the iterations still converge to the root in different situation. I will recommend sketched graph method.

### Solution c

Seriously, i wont suggest students to write this working in STPM exam. Even the F'(x) is more than 1, where x is the first estimate, the iterations still converge to the root in different situation. I will recommend sketched graph method.

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