# Thread: Infinitesimal 2. Second question!

1. ## 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!

2. 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 $\displaystyle \tan(x)$. There is such a zero at $\displaystyle x=\pi$ which is close to your starting value, and a little experimentation indicated that the sequence converges to this zero of $\displaystyle \tan(x)$. Now you just have to prove that it converges.

You will need to use a result something like:

$\displaystyle |\varepsilon|\le |\tan(\pi+\varepsilon)|<|2\varepsilon|$

for $\displaystyle |\varepsilon|<0.5$

CB