limit of (1+sn)n is one when limit of nsn is zero

The inequalities for differences of powers may be used to show that limn(1+sn)n=1 when limnnsn=0. This fact plays an important role in the development of the theory of the exponential functionDlmfDlmfMathworldPlanetmathPlanetmath as a limit of powers.

To derive this limit, we bound 1+sn using the inequalities for differences of powers.


Since limnnsn=0, there must exist N such that nsn<1/2 when n>N. Hence, when n>N,


so, as n, we have (1+sn)n1.

Title limit of (1+sn)n is one when limit of nsn is zero
Canonical name LimitOf1SnnIsOneWhenLimitOfNSnIsZero
Date of creation 2013-03-22 15:48:55
Last modified on 2013-03-22 15:48:55
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Proof
Classification msc 26D99