Determining lim_(n->oo}(n^((1)/(n))-1)^n with only elementary math I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of differentiation or other topics of advanced math, only induction and school math are possible. Thank you

Roselyn Daniel 2022-07-21 Answered
Determining lim n ( n 1 n 1 ) n with only elementary math
I am trying to find this limit:
lim n ( n 1 n 1 ) n ,
I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of differentiation or other topics of advanced math, only induction and school math are possible. Thank you
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Answers (2)

eri1ti0m
Answered 2022-07-22 Author has 11 answers
For n 5 we have, using the binomial theorem
( 1 + 1 2 ) n 1 + n 2 + n ( n 1 ) 8 1 + n 2 + n ( 5 1 ) 8 = 1 + n > n
Thus 1 + 1 2 n 1 / n , or
0 ( n 1 / n 1 ) n 1 2 n
(the first inequality being immediate.) Tking the limit as n tend to , we get
lim n ( n 1 / n 1 ) n = 0.

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anudoneddbv
Answered 2022-07-23 Author has 2 answers
Perhaps the simplest way is to go step by step. First, show that n 1 / n 1 as n . You can then conclude that 0 < n 1 / n 1 < 1 2 for large enough values of n, meaning that, when n , the limit must be 0 (sandwich principle).

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