Dit is een handig overzichtje hoe je de rijen of reeksen kan vinden en de verschillende convergentie tesen voor alle reeksen en rijen (ook machtreeksen, McLaurin ontwikkeling...) gaat over p11-14, 81-120 deel 2 wiskunde
Getallenrij Getallenreeks
∞
{un} = u1, u2, …, un …
∑ un = u 1 + u2 + … + un + … : berekenbaar? reekssom
n =1
n
Gedrag:
Partieelsom: Sn = ∑ u i = u1 + u2 + … + un
o Convergent: nlim
→∞
un bestaat en is eindig (∈ R ) i=1
∞
Convergentie
o Divergent: nlim un = ± ∞ Verband reekssom en getallenreeks : ∑ u n = nlim
→∞
Sn
→∞ n =1
o Onbepaald: nlim
→∞
un bestaat niet Gedrag getallenreeks:
o Convergent: reekssom S = nlim
→∞
Sn ∈ R
o Divergent: reekssom S = nlim
→∞
S n=± ∞
o Onbepaald: reekssom S = nlim
→∞
S n bestaat niet
∞ ∞
Speciale Rekenkundige {un}
Veelvoud getallenreeks: ∑ u n convergent ⇔ ∑ α un convergent (α ∈ R)
getallenrijen getallenrij n =1 n=1
∞ ∞
Deel getallenreeks: ∑ u n convergent ⇔ ∑ α un convergent (k ≥ 1)
n =1 n =k
∞
Algemene term: un = u1 + (n – 1).d
Voorwaarde convergentie: ∑ u n convergent ⟹ nlim
→∞
un=0
n =1
Notatie: d = un – un-1 (n ≥ 2) Speciale Rekenkundige reeks ∞
∑ un
getallenreekse n =1
Gedrag getallenrij : Algemen term: un = u1 + (n – 1).d, n ≥ 1
1
, o Convergent: d = 0, nlim
→∞
un=u 1 n
Partieelsom: Sn =
n
(u1 + un), n ≥ 1
2
o Divergent: d ≠ 0, nlim
→∞
un=±∞
ALTIJD divergent, tenzij elke term = 0
∞
Meetkundige {un} Meetkundige reeks
∑ un
getallenrij n =1
Algeme term: un = u1 . qn-1 Algemene term : un = un . qn-1, n ≥ 1
un Partieelsom :
Notatie : q = (n ≥ 2)
un−1 o q = 1 : Sn = n . u 1
n
o q ≠ 1 : Sn = u1 .
1−q
1−q
Voor n ≥ 1
Gedrag getallenrij : Gedrag getallenreeks:
o Convergent : o Convergent: -1 < q < +1
o -1 < q < +1, nlim
→∞
un=0
o Reekssom: S =
u1
1−q
o q = +1, nlim
→∞
un=u 1
o Divergent : q ≥ 1
o Divergent : q > 1, nlim
→∞
un=±∞ o Onbepaald: q ≤ -1
o Onbepaald : q ≤ -1, nlim
→∞
un bestaat niet
∞
Hypeharmonische {un} Hypeharmonische reeks
∑ un
getallenrij n =1
2
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