Infinite Continued Fractions

In this article we are going to look at infinite continued fractions - continued fractions that do not terminate.

Hyperbolic Thinking

Stage: 5 Challenge Level:

Start by looking at the values of $A^2(x), B^2(x), A(2x), B(2x), A(x)B(x)$ and see if you can spot any simple relationships linking them.

To differentiate the two functions use the expression
$$\frac{d}{dx}\left(a^x\right) = \ln(a) a^x$$

Consider simple values of the functions, such as $n\pi$ and $\pm 1$.

Maclaurin series. Trigonometry generally. Trigonometric identities. Exponential function. Complex numbers. Sine & cosine in terms of complex exponential function. Hyperbolic functions. Iteration. Integration. Hyperbolic trig identities.

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Infinite Continued Fractions

Hyperbolic Thinking

Stage: 5 Challenge Level: