Acel 0,7 ne spune si cat de repede converge sirul. (Avatar in cadrul teoremei de punct fix a lui Banach.) Avem anume pentru $n>1$:
Iata primii 20 de termeni ai sirului...
(19:31) gp > f(x) = 2^(x/2)
(19:31) gp > a=1
%15 = 1
(19:31) gp > for( n=0,20, print( "x[" , n, "] ~ ", a ); a=f(a) )
x[0] ~ 1
x[1] ~ 1.414213562373095048801688724
x[2] ~ 1.632526919438152844773495381
x[3] ~ 1.760839555880028090756649896
x[4] ~ 1.840910869291010298414834541
x[5] ~ 1.892712696828510480823454616
x[6] ~ 1.926999701847100431015638318
x[7] ~ 1.950034773805817581828743270
x[8] ~ 1.965664886517318713890147613
x[9] ~ 1.976341754409702377696340625
x[10] ~ 1.983668399303821147193751746
x[11] ~ 1.988711773413953318328100793
x[12] ~ 1.992190882947057099754305148
x[13] ~ 1.994594450712101177628954135
x[14] ~ 1.996256666265858383796961722
x[15] ~ 1.997407001141335832237269445
x[16] ~ 1.998203477508701623889882901
x[17] ~ 1.998755133084591885137007335
x[18] ~ 1.999137310119390933251325220
x[19] ~ 1.999402118324996572890407661
x[20] ~ 1.999585622935681030440026877
Ce inseamna aplicarea teoremei lui Banach legata de convergenta (si rapiditatea ei) in multe cazuri particulare este un lucru important.
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