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Root test

  1. Jul 20, 2014 #1
    Hello.
    How do I find the limit of this term?

    $$\lim_{{n}\to{\infty}}|\left(\frac{n}{n+1}\right)^{\!{n^2}}|^\frac{1}{n}$$

    This is the working but I don't understand how to get the third line.

    r = lim(n→∞) |[n/(n+1)]^(n^2)|^(1/n)
    ..= lim(n→∞) [n/(n+1)]^n
    ..= lim(n→∞) 1 / [(n+1)/n]^n
    ..= lim(n→∞) 1 / (1 + 1/n)^n
    ..= 1/e, by the limit definition of e.
     
  2. jcsd
  3. Jul 20, 2014 #2

    Erland

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    ##\frac 1{\frac{n+1}n}=\frac n{n+1}## and ##(\frac 1a)^n = \frac 1{a^n}##, right?
     
  4. Jul 20, 2014 #3
    Yes, but as you can see, the third line has its denominator to the power of n. That's what I don't understand. If we divide all terms with n, then what happens to the n outside the bracket?
     
  5. Jul 20, 2014 #4

    Erland

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    Sorry, there was a typo in my reply which I corrected immediatly, but you were so quick and got the typo in your reply...
     
  6. Jul 20, 2014 #5
    Yeah, I was quick, haha. As soon as I submitted my post, I saw you have corrected your error. Thank you!
     
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