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III. Potensrækker – 2. Talfølger og rækker 268 4 Sætning Paspå! lim n→∞ |an| = 0 ⇒ lim n→∞ an = 0 lim n→∞ |(−1)n | = 1 Men talfølgen an = (−1) n er divergent. 2.9. Potenser ☞ [S] 8.1 Sequences 6 Sætning Talfølgen {r n } er konvergent for −1 < r ≤ 1 med Talfølgen er divergent for øvrige r. lim n→∞ rn = 0 hvis − 1 < r < 1 1 hvis r = 1 2.10. Monotone følger ☞ [S] 8.1 Sequences Definition Betragt en talfølge {an}. • voksende: an < an+1, (≤)
III. Potensrækker – 2. Talfølger og rækker 269 • aftagende: an > an+1, (≥) • monoton: voksende eller aftagende • opadtil begrænset: an ≤ M • nedadtil begrænset: m ≤ an • begrænset: opadtil- og nedadtil begrænset 2.11. Monoton og begrænset er konvergent ☞ [S] 8.1 Sequences 7 Sætning Enhver monoton begrœnset talfølge er konvergent. Eksempel Følgen = 1 1 + 1 n an = n n + 1 1 2 3 4 5 , , , , 2 3 4 5 6 ,... er voksende og begrænset. Følgen er dermed konvergent. 2.12. Rekursiv følge ☞ [S] 8.1 Sequences Eksempel 11 Talfølgen {an} a1 = 2, an+1 = 1 2 (an + 6)
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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