limit of geometric sequence

As mentionned in the geometric sequenceMathworldPlanetmath entry,

limnarn=0 (1)

for  |r|<1.  We will prove this for real or complex values of r.

We first remark, that for the values  s>1  we have  limnsn= (cf. limit of real number sequence).  In fact, if M is an arbitrary positive number, the binomial theoremMathworldPlanetmath (or Bernoulli’s inequality) implies that


as soon as  n>Ms-1.

Let now  |r|<1  and ε be an arbitrarily small positive number.  Then  |r|=1s  with s>1.  By the above remark,


when  n>1(s-1)ε.  Hence,


which easily implies (1) for any real number a.

Title limit of geometric sequence
Canonical name LimitOfGeometricSequence
Date of creation 2013-03-22 18:32:43
Last modified on 2013-03-22 18:32:43
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Proof
Classification msc 40-00