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!

Why does the limit(n->∞) sqrt(n)/(sqrt(n)+sqrt(n+1)) equal 1/(1+sqrt(1-1/n))?

  1. Nov 5, 2014 #1
    1. The problem statement, all variables and given/known data
    I am studying for a calculus test tomorrow on this website (http://archives.math.utk.edu/visual.calculus/6/index.html). I am working on the limit comparison test problems but I am unfamiliar with the form they use in their solutions. For example:
    Limit comparison test (prove convergence / divergence)
    Series (from n=1 to ∞) 1/(sqrt(n)+sqrt(n+1))

    2. Relevant equations

    3. The attempt at a solution
    Series (from n=1 to ∞) 1/(sqrt(n)+sqrt(n+1)) compare to Series (from n=1 to ∞) 1/sqrt(n), which we know diverges by the p-series test (p=.5, p<1)
    Let an = 1/(sqrt(n)+sqrt(n+1)) and bn = 1/sqrt(n)
    lim(n->∞) an/bn
    = lim(n->∞) 1/(sqrt(n)+sqrt(n+1)) / 1/sqrt(n)
    = lim(n->∞) sqrt(n)/(sqrt(n)+sqrt(n+1))

    I can see why this limit is 1/2 but I don't understand how the solution to the problem ended up in the form
    lim(n->∞) 1/(1+sqrt(1-1/n)) = 1/2
    I can tell by the other solutions that this form is a rule of sorts but I don't know what it is.
    Thank you for the help.
  2. jcsd
  3. Nov 5, 2014 #2


    Staff: Mentor

    Factor ##\sqrt{n}## from the two terms of ##\sqrt{n}## + ##\sqrt{n + 1}##.
  4. Nov 5, 2014 #3
    OH. so if you multiply
    sqrt(n)/(sqrt(n)+sqrt(n+1)) by (1/sqrt(n))/(1/sqrt(n)) you get 1/(sqrt(n)+sqrt(n+1))/sqrt(n) which can simplify to 1/(sqrt(n/n)+sqrt(n/n+1/n)) or 1/(1+sqrt(1+1/n))
  5. Nov 5, 2014 #4


    Staff: Mentor

    No, it's simpler than that. Just factor ##\sqrt{n}## out of the numerator and denominator. The ##\sqrt{n}## factors can be cancelled, and you're left with the form you see in the solution.
  6. Nov 5, 2014 #5
    Oh, ok. I understand. Thank you!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted