limit of sequence as sum of series


If U is the limit of a sequencePlanetmathPlanetmath

u1,u2,u3,

of real or complex numbersMathworldPlanetmathPlanetmath, then U can be expressed as the series sum

U=u1+i=1(ui+1-ui).

Proof. Let  sn:=u1+i=1n-1(ui+1-ui).  We see that

sn=u1+i=1n-1ui+1-i=1n-1ui=u1+j=2nuj-i=1n-1ui=un

for all  n=1, 2, 3,  Thus

u1+i=1(ui+1-ui)=limnsn=limnun=U,

Q.E.D.

Title limit of sequence as sum of series
Canonical name LimitOfSequenceAsSumOfSeries
Date of creation 2013-03-22 17:28:21
Last modified on 2013-03-22 17:28:21
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 4
Author pahio (2872)
Entry type Theorem
Classification msc 40-00
Related topic SumOfSeries