Let a_{1} > 1, and suppose a_{n+1} = 2 - 1/a_{n} for all n >= 2 . Prove that (a_{n}) converges, and find its limit.
Last edited by Kanwar245; Mar 15th 2013 at 07:43 PM.
Follow Math Help Forum on Facebook and Google+
Originally Posted by Kanwar245 Let a_{1} > 1, and suppose a_{n+1} = 2 - 1/a_{n} for all n >= 2 . Prove that (a_{n}) 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 $\displaystyle L=2-\frac{1}{L}$.
Don't you think there's a typo, since we are given a1 > 1, and a_n+1 is for n>= 2
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
Last edited by Plato; Mar 15th 2013 at 03:46 PM.
Okay, so when proving through induction that it's monotonically decreasing, I just need to prove that a_{n+1} >= a_{n} right
Originally Posted by Kanwar245 Okay, so when proving through induction that it's monotonically decreasing, I just need to prove that a_{n+1} >= a_{n} right BASE CASE: Prove that $\displaystyle a_1>a_2\ge 1$. PROVE Inductive case: If $\displaystyle a_k>a_{k-1}\ge 1$ then $\displaystyle a_{k+1}>a_k\ge 1$.
View Tag Cloud