limit of nth root of n

The nth root ( of n tends to 1 as n tends to infinity, i.e. the real number sequence


converges to the limit

limnnn=1. (1)

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


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


whence  limnδn=0.  Accordingly,



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