# Limit of a strange sequence

• Nov 11th 2012, 07:42 AM
Femto
Limit of a strange sequence
Define sequence by a_0 = 1 and a_(n + 1) = cos(a_(n)/2). Given that |cos(x) - cos(y)| < (or equal to) |x - y| use this to show that cos(a_n/2) -> cos(a/2).

Not exactly sure how to get this. I'm probably being inane here. This question comes under Cauchy sequences..
• Nov 11th 2012, 08:24 AM
Plato
Re: Limit of a strange sequence
Quote:

Originally Posted by Femto
Define sequence by a_0 = 1 and a_(n + 1) = cos(a_(n)/2). Given that |cos(x) - cos(y)| < (or equal to) |x - y| use this to show that cos(a_n) -> cos(a/2).
Not exactly sure how to get this. I'm probably being inane here. This question comes under Cauchy sequences..

Is it $\displaystyle \left(\cos\left(a_n\right)\right)\to\cos\left( \frac{a}{2} \right)~?$

If so, what is $\displaystyle a$ in that?
• Nov 11th 2012, 08:43 AM
Femto
Re: Limit of a strange sequence
Quote:

Originally Posted by Plato
Is it $\displaystyle \left(\cos\left(a_n\right)\right)\to\cos\left( \frac{a}{2} \right)~?$

If so, what is $\displaystyle a$ in that?

No it's not. I've written it wrong - it should be cos(an/2)
• Nov 11th 2012, 08:51 AM
Plato
Re: Limit of a strange sequence
Quote:

Originally Posted by Femto
No it's not. I've written it wrong - it should be cos(an/2)

What should be cos(an/2)? That makes no sense.