The Functional Continued Fraction (F C F) of exponential class is defined by #a_(cf) (x;b) = a^(x+b/(a^(x+b/a^(x +...)))), a > 0#. Upon setting a = e = 2.718281828.., how do you prove that #e_(cf) ( 0.1; 1 ) = 1.880789470#, nearly?
See explanation...
Then:
Then:
So we find:
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Contractive behaviour.
Let us investigate the conditions for a contraction in the iteration operator.
Substracting both sides
but in first approximation
or
To have a contraction we need
This is attained if
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[e_{cf}(0.1; 1) \approx 1.880789470]
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To prove that ( e_{cf}(0.1;1) ) is approximately ( 1.880789470 ) when ( a = e ), you would use the formula for the Functional Continued Fraction (F C F) of exponential class:
[ a_{cf}(x;b) = a^{(x+\frac{b}{a^{(x +\frac{b}{a^{(x +...)}))}}}} ]
Substitute ( a = e ), ( x = 0.1 ), and ( b = 1 ) into the formula and compute the value iteratively until convergence. The result will be approximately ( 1.880789470 ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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