tallandpoofy said:Homework Statement
Let x1 > 9000, and
xn+1 = )2009xn + 2010)/2011 for n >1
show that (xn) converges and find its limit
Homework Equations
Definition of a limit, Monotone Convergence Theorem.
The Attempt at a Solution
Since xn+1 is monotone for n>1 and bounded, then it converges by Monotone Convergence Theorem.
How do i prove monotone for this function?
I tried xn+2 - xn+1 < 0 but it does not work since xn+1 is dependent on xn
A sequence is said to converge if its terms become closer and closer to a single value as the index (or term number) increases. In other words, the terms of the sequence eventually approach a specific limit.
To prove convergence of a sequence, you must show that the terms of the sequence get arbitrarily close to a specific limit value. This can be done by using a variety of techniques such as the squeeze theorem, the ratio test, or the root test.
A sequence dependent on previous terms is a sequence where each term is determined by the previous terms in the sequence. This means that the value of each term is not independent and is affected by the values of the previous terms.
Proving convergence of a sequence dependent on previous terms is important because it allows us to understand the behavior of the sequence and determine if it will eventually approach a specific limit. This can have applications in various fields such as physics, engineering, and economics.
Some common techniques used to prove convergence of a sequence dependent on previous terms include the monotone convergence theorem, the Cauchy criterion, and the Bolzano-Weierstrass theorem. These techniques involve analyzing the behavior and properties of the sequence to determine if it converges to a specific limit.