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Discussion –

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Discussion –

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# How to identify an iteration which is converges?

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 $x=\alpha$ is an approximation to a root of the equation $x=F(x)$, then sequence defined by $x_{n+1}=F(x_n)$ with a starting value close to $\alpha$ will converge to $\alpha$ as long as $|F'(\alpha)|<1$.

## The Real

If $x=\alpha$ is a real root of the equation $x=F(x)$, then sequence defined by $x_{n+1}=F(x_n)$ with a starting value close to $\alpha$ will converge to $\alpha$ as long as $|F'(\alpha)|<1$.

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

### 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.

### Solution d

#### KK LEE

KK LEE has been a STPM Mathematics tuition teacher since June 2006, but his love of Maths dates back to at least 1999 when he was Form 4. KK LEE started teaching in 2006 at Pusat Tuisyen Kasturi. He was known as "LK" when he was teaching in PTK. After teaching STPM Mathematics for 8 years in PTK, he joined Ai Tuition Centre in 2014. Over the years he has taught Mathematics (T), Mathematics (S), Mathematics (M), Additional Mathematics.

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