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| 2, X2 = 1.75,
X2 = 1.732142857, X4 =
1.73205081 |
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| 1.732050808, X6
= 1.732050808 , X7 1.732050808 |
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We notice that when Xn = 1.732050808, so
is Xn+1. Squaring these terms we get
X12 = 4,
X22 = 3.0625, ... ,
X52 = 3 and the rest of the other
terms are the same!!
This implies that when Xn » Ö3 so is
Xn+1 and the values of
Xn tend to the limit Ö3. This special property can easily be
proven. Assume that the limit exists, so
Xn+1 = Xn =
X, then solve the equation
| X = |
1
2
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æ
ç
è |
X + |
3
N
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ö
÷
ø |
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If we test it for N = 3, we see that
X29 = 1.44224957, which is what the calculator
gives for the cube root of 3. Testing it for N = 8, we get
X1 = 2, which is the right answer.