As a small introduction to continued fractions, often used for finding rational approximations to irrational constants, in this video we build one for √z (principle branch) using a difference of squares factorization of z - 1 and find the range of convergence for complex inputs z (considering only real inputs goes through the same process and analysis). Lots of parallels can be drawn to infinite series.
Timecodes:
0:00 - Finding the continued fraction
3:17 - Outline of convergence for continued fractions
9:08 - Determining convergence
18:18 - Special case of z = 0
Addendum: If you want negative real roots, just take the positive root and multiply by i in the formula, also you can multiply by -1 to get the other branch of the square root
Sources and other tidbits:
Continued Fraction: mathworld.wolfram.com/Continu...
Generalized Continued Fraction: en.wikipedia.org/wiki/General...
Approximations in Desmos: www.desmos.com/calculator/zvs...
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