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Limit problem

  1. Nov 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Evaluate [tex] lim_{n→∞} \frac{ (n+1)^{\frac{1}{n+1}} }{n^{\frac{1}{n}}} [/tex]

    2. Relevant equations

    3. The attempt at a solution
    This is actually a part of a series problem I am trying to solve using the ratio test.I can't seem to figure out this limit and L'Hopital's doesn't work. Any hints?

  2. jcsd
  3. Nov 11, 2012 #2


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    Note that if [itex]\lim_{n \rightarrow \infty} n^{1/n}[/itex] exists, let's call it [itex]L[/itex], then both the numerator and denominator converge to [itex]L[/itex], so the limit of the fraction is 1.

    Indeed, [itex]\lim_{n \rightarrow \infty} n^{1/n}[/itex] does exist, so focus on proving that fact.
  4. Nov 11, 2012 #3
    hey i think the limit is 1

    1/(n+1) → 0
    1/n → 0

    if both powers will go → 0 then everything0 = 1

    then you get 1/1 = 1
  5. Nov 11, 2012 #4


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    The answer is right, but the argument is invalid. It's true that [itex]x^{1/n} \rightarrow 1[/itex] if [itex]x[/itex] is a fixed positive number, but here we have [itex]x[/itex] growing to infinity while the exponent shrinks to zero. It is not automatically true that the limit will be 1.

    Consider for example
    [tex]\lim_{n \rightarrow \infty} (n^n)^{1/n}[/tex]
    Surely this does not converge to 1, since [itex](n^n)^{1/n} = n[/itex].
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