# Sequences convergence problem

• Mar 15th 2013, 04:13 PM
Kanwar245
Sequence convergence problem
Let a1 > 1, and suppose an+1 = 2 - 1/an for all n >= 2 . Prove that (an) converges, and find its limit.
• Mar 15th 2013, 04:33 PM
Plato
Re: Sequences convergence problem
Quote:

Originally Posted by Kanwar245
Let a1 > 1, and suppose an+1 = 2 - 1/an for all n >= 2 . Prove that (an) converges, and find its limit.

It is sufficient to show that the sequence is decreasing and bounded below.
Do that with induction.

The solve the equation $L=2-\frac{1}{L}$.
• Mar 15th 2013, 04:36 PM
Kanwar245
Re: Sequences convergence problem
Don't you think there's a typo, since we are given a1 > 1, and a_n+1 is for n>= 2
• Mar 15th 2013, 04:40 PM
Plato
Re: Sequences convergence problem
Quote:

Originally Posted by Kanwar245
Don't you think there's a typo, since we are given a1 > 1, and a_n+1 is for n>= 2

Why should anyone think that?

Take a look at this
• Mar 15th 2013, 04:47 PM
Kanwar245
Re: Sequences convergence problem
Okay, so when proving through induction that it's monotonically decreasing, I just need to prove that an+1 >= an right
• Mar 15th 2013, 05:12 PM
Plato
Re: Sequences convergence problem
Quote:

Originally Posted by Kanwar245
Okay, so when proving through induction that it's monotonically decreasing, I just need to prove that an+1 >= an right

BASE CASE: Prove that $a_1>a_2\ge 1$.

PROVE
Inductive case: If $a_k>a_{k-1}\ge 1$ then $a_{k+1}>a_k\ge 1$.