Thread: Infinitesimal 2. Second question!

Given x_0 = 3, we define recursively (x_(n)), an infinite sequence starting at n=1 and the recursive formula is:
x_(n+1) = x_(n) - tan x_(n)

Prove that the sequence is convergent and find her limit.

* You may use the fact that: |tan3|<0.1426



Thank you all!

Originally Posted by Also sprach Zarathustra

Given x_0 = 3, we define recursively (x_(n)), an infinite sequence starting at n=1 and the recursive formula is:
x_(n+1) = x_(n) - tan x_(n)

Prove that the sequence is convergent and find her limit.

* You may use the fact that: |tan3|<0.1426



Thank you all!

If this converges it converges to a zero of . There is such a zero at which is close to your starting value, and a little experimentation indicated that the sequence converges to this zero of . Now you just have to prove that it converges.

You will need to use a result something like:



for

CB