MATEMATIKA 1 skripta studij: Biotehnologija i Prehrambena ... - PBF
MATEMATIKA 1 skripta studij: Biotehnologija i Prehrambena ... - PBF
MATEMATIKA 1 skripta studij: Biotehnologija i Prehrambena ... - PBF
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Rješenje: Vrijedi: s n = a 1 · 1−qn . Primijenimo to:<br />
1−q<br />
1+ 1<br />
lim<br />
+ 1 +...+ 1<br />
3 3 2 3 n<br />
n→∞ 1+ 1 + 1 +...+ 1 = lim<br />
5 5 2 5 n<br />
1· 1−(1/3)n+1<br />
1−1/3<br />
n→∞ 1· 1−(1/5)n+1<br />
1−1/5<br />
= 6 5<br />
Zadatak 51<br />
a n = ( √ n+2− √ n+1) cosn<br />
Rješenje:<br />
lim (√ n+2− √ √ √ n+2+ n+1<br />
n+1)· √ √ ·cosn<br />
n→∞ n+2+ n+1<br />
1<br />
= lim √ √ ·cosn = 0<br />
n→∞ n+2+ n+1<br />
budući je cos ograničena funkcija.<br />
Zadatak 52<br />
Rješenje:<br />
a n = 1 n 2 + 2 n 2 +...+ n n 2<br />
1+2+...+n<br />
lim<br />
n→∞ n 2<br />
= lim<br />
n(n+1)<br />
2<br />
n→∞ n 2<br />
= 1 2 lim 1+ 1 n<br />
n→∞ 1<br />
= 1 2<br />
NEODREDENI OBLICI:<br />
0<br />
0 , ∞<br />
∞ , ∞−∞, 0·∞, ∞0 , 1 ∞ , 0 0<br />
Zadatak 53<br />
a) a n = n ( √ n 2 +1+ √ n 2 −1), b) a n = n ( √ n 2 +1− √ n 2 −1)<br />
40
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