Binomialrekken
Binomialrekken
Binomialrekken
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<strong>Binomialrekken</strong><br />
<strong>Binomialrekken</strong> er Taylorrekken rundt x = 0 til funksjonen<br />
f (x) = (1 + x) m ,<br />
der m er en konstant
<strong>Binomialrekken</strong><br />
<strong>Binomialrekken</strong> er Taylorrekken rundt x = 0 til funksjonen<br />
f (x) = (1 + x) m ,<br />
der m er en konstant<br />
Viser seg å bli (se bok):<br />
<strong>Binomialrekken</strong><br />
(1 + x) m =<br />
( )<br />
∞∑ m<br />
x n , |x| < 1<br />
n<br />
der ( )<br />
m m!<br />
=<br />
, dersom m ≥ 0.<br />
n n!(m − n)!<br />
Generelt:<br />
n=0
<strong>Binomialrekken</strong><br />
<strong>Binomialrekken</strong> er Taylorrekken rundt x = 0 til funksjonen<br />
f (x) = (1 + x) m ,<br />
der m er en konstant<br />
Viser seg å bli (se bok):<br />
<strong>Binomialrekken</strong><br />
(1 + x) m =<br />
( )<br />
∞∑ m<br />
x n , |x| < 1<br />
n<br />
der ( )<br />
m m!<br />
=<br />
, dersom m ≥ 0.<br />
n n!(m − n)!<br />
Generelt:<br />
( )<br />
m<br />
= 1,<br />
0<br />
( )<br />
m<br />
= m,<br />
1<br />
n=0<br />
( )<br />
m<br />
=<br />
n<br />
m(m − 1)(m − 2) · · · (m − n + 1)<br />
n!
<strong>Binomialrekken</strong> – Eksempel<br />
( )<br />
1<br />
∞∑<br />
1 + x = (1 − −1<br />
x)−1 = x n , |x| < 1<br />
n<br />
n=0<br />
( ) ( ) ( )<br />
−1 −1 −1<br />
= + x + x 2 + · · ·<br />
0 1 2
<strong>Binomialrekken</strong> – Eksempel<br />
Har<br />
( )<br />
1<br />
∞∑<br />
1 + x = (1 − −1<br />
x)−1 = x n , |x| < 1<br />
n<br />
n=0<br />
( ) ( ) ( )<br />
−1 −1 −1<br />
= + x + x 2 + · · ·<br />
0 1 2<br />
( )<br />
−1<br />
=<br />
n<br />
( )<br />
−1<br />
= 1,<br />
0<br />
( )<br />
−1<br />
= −1<br />
1<br />
(−1)(−1 − 1)(−1 − 2) · · · (−1 − n + 1)<br />
n!<br />
= (−1) n n!<br />
n! = (−1)n
<strong>Binomialrekken</strong> – Eksempel<br />
Har<br />
Dermed<br />
( )<br />
1<br />
∞∑<br />
1 + x = (1 − −1<br />
x)−1 = x n , |x| < 1<br />
n<br />
n=0<br />
( ) ( ) ( )<br />
−1 −1 −1<br />
= + x + x 2 + · · ·<br />
0 1 2<br />
( )<br />
−1<br />
=<br />
n<br />
( )<br />
−1<br />
= 1,<br />
0<br />
( )<br />
−1<br />
= −1<br />
1<br />
(−1)(−1 − 1)(−1 − 2) · · · (−1 − n + 1)<br />
n!<br />
= (−1) n n!<br />
n! = (−1)n<br />
∞<br />
1<br />
1 + x = ∑<br />
(−1) n x n .<br />
n=0
Mye brukte rekker<br />
1<br />
1 − x = ∑<br />
x n ,<br />
n=0<br />
|x| < 1<br />
∞∑<br />
e x x n<br />
=<br />
n=0 n! , |x| < ∞<br />
∞∑ (−1) n<br />
sin x =<br />
n=0 (2n + 1)! x2n+1 , |x| < ∞<br />
∞∑ (−1) n<br />
cos x =<br />
n=0 (2n)! x2n , |x| < ∞<br />
∞∑ (−1) n−1<br />
ln(1 + x) =<br />
x n ,<br />
n=1 n<br />
1 < x ≤ 1<br />
∞∑<br />
( )<br />
(1 + x) m m<br />
= x n ,<br />
n=0 n<br />
|x| < 1<br />
∞∑<br />
tan −1 (−1) n<br />
x =<br />
n=0 2n + 1 x2n+1 , |x| ≤ 1