limit of nth root of n


The nth root (http://planetmath.org/NthRoot) of n tends to 1 as n tends to infinity, i.e. the real number sequence

11,22,33,,nn,

converges to the limit

limnnn=1. (1)

Proof.  If we denote  nn:=1+δn, we may write by the binomial theorem that

n=(1+δn)n=1+(n1)δn+(n2)δn2++(nn)δnn.

This implies, since all hand side are positive (when  n>1), that

n>(n2)δn2=n(n-1)2!δn2,δn2<2n-1,0<δn<2n-1,

whence  limnδn=0.  Accordingly,

limnnn=limn(1+δn)=1,

Q.E.D.

Note.  (1) follows also from the corollary 3 in the entry growth of exponential function.

Title limit of nth root of n
Canonical name LimitOfNthRootOfN
Date of creation 2014-09-28 13:20:59
Last modified on 2014-09-28 13:20:59
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Example
Classification msc 12D99
Classification msc 30-00
Synonym sequence of nth roots of n