Numerikus sorok - Index of
Numerikus sorok - Index of
Numerikus sorok - Index of
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Ha<br />
Ha<br />
∞∑<br />
bn konvergens, akkor a n < c 2 b n miatt<br />
∞∑<br />
bn divergens, akkor c 1 b n < a n miatt<br />
∞∑<br />
an is konvergens (majoráns kritérium)<br />
∞∑<br />
an is divergens (minoráns kritérium)<br />
✓✏∞∑ ∞<br />
1<br />
Pl.<br />
✒✑ 3n 2 − 2 √ n + 8 = ∑<br />
an<br />
a n ∼ 1<br />
3n 2 = b n<br />
és<br />
✓✏∞∑ ∞<br />
1<br />
Pl.<br />
✒✑ 7n + 3√ n − 1 = ∑<br />
an<br />
a n ∼ 1<br />
7n = b n<br />
és<br />
✓✏∞∑ ∞<br />
1<br />
Pl.<br />
✒✑ arctg<br />
n = ∑<br />
an<br />
∞∑<br />
bn konvergens =⇒<br />
∞∑<br />
bn divergens =⇒<br />
∞∑<br />
an konvergens<br />
∞∑<br />
an divergens<br />
a n ∼ 1 n = b n<br />
és<br />
∞∑<br />
bn divergens =⇒<br />
∞∑<br />
an divergens<br />
✓✏∞∑ (<br />
Pl.<br />
✒✑ 1 − cos<br />
n)<br />
1 =<br />
∞∑<br />
an<br />
a n = 2 sin 2 1<br />
2n ∼ 2 1<br />
4n 2 = 1<br />
2n 2 = b n<br />
és<br />
∞∑<br />
bn konvergens =⇒<br />
∞∑<br />
an konvergens<br />
Feladatok:<br />
Konvergensek-e az alábbi <strong>sorok</strong>?<br />
∞∑<br />
( 1<br />
1.<br />
n n)<br />
− arctg 1<br />
2.<br />
n=1<br />
∞∑<br />
n=1<br />
(<br />
ch 2 n − cos 3 )<br />
n<br />
3.<br />
∞∑<br />
n=1<br />
1<br />
( 3n<br />
n<br />
)<br />
c○ Kónya I. – Fritz Jné – Győri S. 37 v1.4