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Beispiel Zu zeichnen sind die WEB-Grafiken für die Rekursionsformeln
an+1 = –3(an)2 + 3an mit a0 = 0,01 und bn+1 = 3bn + 0,2 mit b0 = 0,11.
Die so definierten Zahlenfolgen sind auf Divergenz bzw. Konvergenz
zu untersuchen. Verwenden Sie den folgenden Tabellenindexbereich
und die folgenden Betrachtungsfenster-Einstellungen.
Tabellenindexbereich: Startindex = 0, Endindex = 6,
Cursor-Start bei anStr = 0,01 bzw. bnStr = 0,11 auf der x-Achse.
Betrachtungsfenster-Einstellungen
Xmin = 0, Xmax = 1, Xscale = 1
Ymin = 0, Ymax = 1, Yscale = 1
Vorgang
1m RECUR
2!K(V-Window)awbwbwc
awbwbwi
33(TYPE)c(an+1=)-d2(an)x+d2(an)w
d3(bn)+a.cw
45(RANG)1(a0)
awgwa.abwa.bbwc
a.abwa.bbwi
56(TABL)
64(WEB)
71(TRACE)w~w(für (an) gilt Konvergenz)
cw~w(für (bn) gilt Divergenz)
Ergebnisanzeige
5-9-8
Grafische Darstellung von Rekursionsformeln
Interpretation: Der Fixpunkt a der positiven Zahlenfolge (an) lautet a = -3a2+3a,
d.h
.
a= 2/3.
Damit ist die positive Zahlenfolge (an) konvergent.
Die positive Zahlenfolge (bn) besitzt keinen positiven Fixpunkt (Divergenz).
219


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