Finding continued fractions from algebraic manipulations can be hard, but relating them to general forms of infinite series allows for a wider class of representations for irrational constants by using specific values of Taylor series for example. The formula covered in this video can even work for non-alternating series with some small transformations.
Timecodes:
0:00 - Preliminary: Arctangent subtraction identity
2:05 - Deriving the continued fraction formula
11:43 - Examples using the formula
15:43 - Extra: Non-alternating version of formula
17:32 - Example (non-alternating)
Sources and other tidbits:
Continued Fraction: https://mathworld.wolfram.com/ContinuedFraction.html
Generalized Continued Fraction: https://en.wikipedia.org/wiki/Generalized_continued_fraction
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