
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
May 9th, 2015, 05:14 AM  #1 
Member Joined: Oct 2013 Posts: 36 Thanks: 0  limit of a recursive sequence
let (a_n) be a sequence such that: a_(n+2)=1/3*a_(n+1)+2/3*a_n what is the limit of (a_n)? [the fact it converges has already been proved earlier in the question and we are not given a(1) and a(2)] 
May 9th, 2015, 05:22 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 
The general solution is$$A\lambda_1^n + B\lambda_2^n$$ where $\lambda_1$ and $\lambda_2$ are roots of the characteristic equation $u^2  \frac13u  \frac23=0$. You probably don't need to know the constants $A$ and $B$. 
May 9th, 2015, 06:55 AM  #3 
Member Joined: Oct 2013 Posts: 36 Thanks: 0 
how did u get this? (a derivation would be greatly appreciated but a brief explanation would also be very helpful if u dont mind)

May 9th, 2015, 07:10 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 
I haven't seen a constructive proof, but a first order difference equation $a_{n+1} = ka_n$ has solution $a_0 k^n$ which could motivate us to seek solutions of the form $a_n = Ar^n$ to the second order equation $a_{n+2}+ba_{n+1}+ca_n=0$. Substituting that into the original equation and solving for $r$ gives the result that $r$ must be a root of the characteristic equation $u^2 + bu + c = 0$. Since this equation has two roots, there are two independent solutions and any linear combination of them will also be a solution because the equation is linear. One ought to prove that such a construction provides all solutions to the equation, but I've never seen such a proof. 
May 9th, 2015, 07:21 AM  #5 
Member Joined: Oct 2013 Posts: 36 Thanks: 0 
the thing is this is from an analysis paper (the syllabus is a bit older but we have definitely not touched on difference equations yet, thats in another course altogether) so is there another way you could see to derive this using only 1st year analysis techniques?

May 9th, 2015, 09:33 AM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 
It seems an odd question to give to someone who hasn't studied the subject at all!

May 9th, 2015, 09:41 AM  #7  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Hello, fromage! Quote:
 

Tags 
limit, recursive, sequence 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Recursive Sequence Question  Becarev  Real Analysis  1  February 11th, 2012 08:17 AM 
limit of recursive sequence  yo12321  Calculus  2  December 8th, 2011 06:23 AM 
Solve Recursive Sequence  milagros  Applied Math  2  November 2nd, 2011 02:44 AM 
RECURSIVE SEQUENCE  milagros  Real Analysis  0  October 26th, 2011 11:46 AM 
recursive forumula for the following sequence  supernova1203  Real Analysis  2  May 18th, 2011 08:36 AM 