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1.30 Potenzreihen
Funktion Potenzreihe
Mathematik
Potenzreihen
Konvergenzbereich
(1 � x) n ⎛n⎞ ⎛n⎞ ⎛n⎞ 2 3
= 1 ± ⎜ ⎟x+ ⎜ ⎟x ± ⎜ ⎟x
+± ... (n beliebig) |x| � 1
⎝1⎠ ⎝2⎠ ⎝3⎠ (1 � x) 1/2 1 1⋅1 ⋅ ⋅ ⋅ ⋅ ⋅
= ± − 2 1 1 3
± 3 1 1 3 5
1 x x x − x 4 ±−...
2 2⋅4 2⋅4⋅6 2⋅4⋅6⋅8 1 1
= ± − 2 1
± 3 5
1 x x x − x4
±−...
2 8 16 128
|x| � 1
(1 � x) 1/3 1 1⋅2 ⋅ ⋅ ⋅ ⋅ ⋅
= ± − 2 1 2 5
± 3 1 2 5 8
1 x x x − x 4±−...
3 3⋅6 3⋅6⋅9 3⋅6⋅9⋅12 1 1
= ± − 2 5
± 3 10
1 x x x − x4±−...
3 9 81 243
|x| � 1
(1 � x) 1/4 1 3
= ± − 2 7
± 3 77
− 4 231
1 x x x x ± x 5 −± ...
4 32 128 2048 8192
|x| � 1
1
(1 ± ) n
n n( n+ 1) + +
= � + 2 n( n 1)( n 2)
1 x x � x 3+
�...
x 1 1⋅2 1⋅2⋅3 |x| � 1
± 1/ 2
1 1 1⋅3 ⋅ ⋅ ⋅ ⋅ ⋅
= � + 2 1 3 5
� 3 1 3 5 7
1 x x x + x 4 �+
...
|x| � 1
(1 x)
2 2⋅4 2⋅4⋅6 2⋅4⋅6⋅8 ± 1/ 3
1 1 1⋅4 ⋅ ⋅ ⋅ ⋅ ⋅
= � + 2 1 4 7
� 3 1 4 7 10
1 x x x + x 4 �+
...
|x| � 1
(1 x)
3 3⋅6 3⋅6⋅9 3⋅6⋅9⋅12 ± 1/ 4
1 1 1⋅5 ⋅ ⋅ ⋅ ⋅ ⋅
= � + 2 1 5 9
� 3 1 5 9 13
1 x x x + x 4 �+
...
|x| � 1
(1 x)
4 4⋅8 4⋅8⋅12 4⋅8⋅12⋅16 1
(1 ± x )
± 2
1
(1 x)
± 3
1
(1 x)
2 3 4
= 1 � x+ x � x + x �+
...
|x| � 1
2 3 4
= 1� 2x+ 3x � 4x + 5 x �+
...
|x| � 1
1
= 1 � (2⋅3x �3⋅ 4x2+ 4⋅5x3 � 5⋅ 6 x 4 + �...
) |x| � 1
2
2 3 4
a x x
= + + 2 x
+ 3 x
1 ln (ln ) (ln ) + (ln ) 4 x
a a a a + ...
|x| � ��
1! 2! 3! 4!
2 3 4
e x x x x x
= 1 + + + + + ...; daraus e:
1! 2! 3! 4!
e
|x| � ��
1 1 1 1
= 1 + + + + ... = 2,718281828459
1! 2! 3!
2 3 4
e –x x x x x
= 1 − + − + −+ ...;
1! 2! 3! 4!
1 1 1 1
1 − + − + −+ ... = 0,367879441
1! 2! 3! 4!
e i x x x x x x
= cos x+ i sin x = 1+ i − − i + + i −−++ ...
1! 2! 3! 4! 5!
daraus e –1 : |x| � ��
2 3 4 5
2 3 4
e – i x x x x x
= cos x− i sin x = 1−i − + i + −−++ ...
1! 2! 3! 4!
Formeln
von Euler
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