1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Proving convergence of recursive sequence

  1. Jan 15, 2008 #1
    1. The problem statement, all variables and given/known data

    A sequence is defined recursively by the equations A1 = 1, An+1 = 1/3(An + 4). Show that {An} is increasing and An < 2 for all n. Deduce that {An} is convergent and find its limit.

    2. Relevant equations

    3. The attempt at a solution

    i've put what i've done in this image.
    http://img297.imageshack.us/img297/8858/62530295kc7.png [Broken]
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Jan 15, 2008 #2
    for n=1 the statement is true
    now suppose it's true for a certain n
    then An+1 = ...<...=2
    here I used the idea that an<2

    Now suppose An+1>An for some n. Use: 1/3(An+4) > An

    Now for n+1, An+2=1/3(An+1 + 4)=1/3(... + 4)=... > 1/3(An+4) if and only if (solve this for An and come to a trivial solution, in example, an<2)

    so now it's increasing and smaller than 2, so...
    For the limit, say an+1=an and solve.
  4. Jan 15, 2008 #3


    User Avatar
    Science Advisor

    That last statement, "for the limit, say An+1= An and solve" is "shorthand" for what really happens and might be misunderstood (obviously, An+1 is never equal to An). If [itex]\alpha[/itex] is the limit (of course, you must have first shown that the limit exists), taking the limit of both sides of the equation, [itex]A_{n+1}= (1/3)(A_n+ 4)[/itex] to get [itex]lim A_{n+1}= (1/3)(lim A_n+ 4)[/itex] which gives [itex]\alpha= (1/3)(\alpha
  5. Jan 15, 2008 #4
    ofcourse, Halls is right. an+1 is not ever an, but they have the same limit as n becomes really big.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook