Key Taylor Series

Single-variable calculus from first principles

A handful of Taylor series come up so often that they're worth knowing cold. Recognising them lets you expand, approximate, and simplify on sight, with no re-deriving the coefficients each time.

Notice the patterns: eˣ uses every power over a factorial; sin uses only odd powers (it's an odd function) and cos only even ones; the geometric series 1/(1−x) is just all powers with coefficient 1.

A series only equals its function within a radius of convergence. For eˣ, sin, and cos the radius is infinite; they work for every x. But 1/(1−x) and ln(1+x) only converge for |x| < 1; push past that and the series diverges into nonsense.

Where this lives in MLThese series are the closed-form backbone of countless ML derivations. The softmax and log-sum-exp rest on the eˣ series; the geometric series 1/(1−γ) gives the value of an infinite discounted reward stream in reinforcement learning; and ln(1+x) appears in log-likelihoods and in stable implementations like log1p. Recognizing the series is how you simplify these expressions by hand.

▶ Key Taylor Series

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