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: Some proofs of convergence

  1. Oct 16, 2008 #1
    1. The problem statement, all variables and given/known data
    1. Prove that the sequence sqrt(n+1) - sqrt(n) converges to 0.
    2. If sequence {an} is composed of real numbers and if lim as n goes to infinity of {a2n} = A and the limit as n goes to infinity of {a(2n-1)} = A, prove that {an} converges to 1. Is converse true?
    3. Consider sequences {an} and {bn}, where bn = (an)^(1/n)
    a. If {bn} converges to 1, does the sequence {an} necessarily converge?
    b. If {bn} converges to 1, does the sequence {an} necessarily diverge?
    c. does {bn} have to converge 1?

    2. Relevant equations

    3. The attempt at a solution
    I'm not sure if I can divide sqrt(n) by sqrt(n) and prove that this new sequence goes to 1 without a loss of generality. As for the others, I am new to these proofs and any help would be much appreciated.
  2. jcsd
  3. Oct 16, 2008 #2


    Staff: Mentor

    1. Certainly you can divide sqrt(n) by itself, as long as n is not 0, but why would you want to do this? Even if you did want to do this, it would be trivial to prove that the limit of that sequence {sqrt(n)/sqrt(n)} is 1.

    Instead, what about multiplying the numerator and denominator by sqrt(n+1) + sqrt(n)? You'd be multiplying by 1, so this won't change the value of the terms in the sequence.

    2. If all the even-subscript terms in the sequence are approaching A, and the odd-subscript terms are doing the same thing, you're going to have a difficult time proving the sequence converges to 1.
  4. Oct 17, 2008 #3


    User Avatar
    Science Advisor

    There is no reason to do that. As Mark44 said, multiply "numerator and denominator" by sqrt{n+1}+ sqrt{n}.

    For problem two are you sure it didn't say "prove that {an} converges to A"? That would make a lot more sense.
    Last edited by a moderator: Oct 23, 2008
  5. Oct 23, 2008 #4
    Wouldn't you just use the def of convergence to prove No. 1
  6. Oct 23, 2008 #5
    need a little help with Xn= (cos n)/(n^3-n^2) and what it converges to.
  7. Oct 23, 2008 #6


    User Avatar
    Science Advisor

    Do NOT add new problems to someone else's threads. Start your own thread.

    Here's a hint: [itex]-1\le cos(n)\le 1[/itex]. Of course, you are assuming n> 1.
  8. Oct 23, 2008 #7
    Sorry I am a newbie!!!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook