Numerikus sorok - Index of
Numerikus sorok - Index of
Numerikus sorok - Index of
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s 2k+1 = 1 → 1<br />
s 2k = 0 → 0<br />
}<br />
=⇒ (s n ) -nek 2 torlódási pontja van, a sor divergens.<br />
✓✏<br />
Pl.<br />
✒✑<br />
∞∑<br />
( ) k 1<br />
= 1 ( ) 2 ( n<br />
(<br />
1 1<br />
2 2 + 1 1<br />
) n<br />
2 − 1<br />
+ · · · + + · · · = lim<br />
2<br />
2)<br />
n→∞<br />
1<br />
2<br />
k=1 } {{ }<br />
− 1 = 1 2<br />
2<br />
} {{ }<br />
s n<br />
s n<br />
−1<br />
− 1 2<br />
= 1 ,<br />
tehát a sor konvergens.<br />
✓✏<br />
Pl.<br />
✒✑<br />
∞∑ 1<br />
k (k + 1) = 1 , mert<br />
k=1<br />
lim<br />
n→∞<br />
n∑<br />
k=1<br />
1<br />
k (k + 1) = lim<br />
n→∞<br />
(( ) −1<br />
= lim<br />
n→∞ 2 + 1 +<br />
(<br />
= lim 1 − 1<br />
n→∞ n + 1<br />
(<br />
n∑ −1<br />
k=1 k + 1 + 1 )<br />
=<br />
k<br />
( −1<br />
3 + 1 ) ( −1<br />
+<br />
2 4 + 1 ) ( −1<br />
+ · · · +<br />
3<br />
n + 1 + 1 ))<br />
=<br />
n<br />
)<br />
= 1, konvergens a sor.<br />
✓✏∞∑<br />
1<br />
Pl.<br />
✒✑ k<br />
k=1<br />
(harmonikus sor) divergens<br />
Ugyanis<br />
( ( 1 1<br />
s 2 k = 1+ +<br />
2)<br />
3 4)<br />
+ 1 ( 1<br />
+<br />
5 + 1 6 + 1 7 + 1 (<br />
+· · ·+ · · · +<br />
8)<br />
1 )<br />
+<br />
2 k−1<br />
≥ 1 + 1 2 + 2 · 1<br />
4 + 4 · 1<br />
8 + · · · + 2k−1 · 1<br />
2 k = 1 + k · 1<br />
2 → ∞<br />
lim s 2 = ∞ =⇒ ∑ ∞<br />
1<br />
k→∞ k k = ∞<br />
Ugyanis s n ≥ s 2 k, ha n > 2 k miatt lim<br />
n→∞<br />
s n = ∞.<br />
•••<br />
k=1<br />
(<br />
1<br />
2 k−1 + 1 + · · · + 1 )<br />
≥<br />
2 k<br />
c○ Kónya I. – Fritz Jné – Győri S. 2 v1.4