Continued Fraction notes

Euclid's Algorithm

Sequences from CF's

Generaized Linear CF's

a x^{2} + b x + c = 0

Hunting divergent CF's

a_{n + 1} = \frac{3 (1 + a_{n})}{3 + a_{n}}

Fibonacci Continued Fraction

Almost Isosceles Right Angled Triangles

n = \frac{1}{1 - n + \frac{2}{2 - n + \frac{3}{3 - n + ...}}}

Ramanujan and an interesting street number

1 + 2 + ... + (n - 1) = (n + 1) + (n + 2) + ... + m

Recursive seqence {1/(an+b)}

Eulers Identity

Alternating sum expressed as a cf

Trigonomentric Functions Table

The continued fractions expansions.

Product and Addition of cf's

Closure.

Converting Power Series to Contined Fractions

Issues and problem generalization.

Convergence of Periodic cfs

Algebra simplification for faster convergence.

Algebraic and Visual Representation of cfs

Algebra of continued fractions and circuits.

\frac{y_{n + 1}}{y_{n}} = y_{n} - y_{n - 1}

Not a cf, but an interesting number.

Challenges

Find the CF expansion of $\log_{b} n$ .
Find the CF for $\int_{0}^{x} e^{- x^{2}} dx$ . Derive it.
Find the continued fraction associated with $s (n) = \frac{n (n + 1)}{2}$ .
Can cf's be used to calculate the floor and ceiling functions? Reference to page 170 of Niven and Zucherman in "An Introduction to The Theory of Numbers".