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Taylor's Theorem — by hand

A reading series · anthology (Algebra)

Working a classic anthology on Taylor's theorem the long way: from binomial coefficients and the metric/topology axioms up to the Taylor series of sin, cos, and exp, then out to q-series and Hensel's number-field/function-field analogy. The handwritten pages live in the Notebook; the polished derivations land on /derivations.

0 of 7 chapters written · a chapter a day.

  1. 1q-series and the q-Pochhammer symbol(a;q)ₙ, infinite products, basic hypergeometric seriesplanned
  2. 2Hensel's analogy — ℤ and ℂ[X]primes vs linear factors; unique factorization; Taylor expansion of a polynomialplanned
  3. 3The binomial theorem(x+y)ⁿ, the coefficients, and Σ(n choose k)=2ⁿplanned
  4. 4Limits, continuity, and the metric axiomsboundary points, continuity, the four conditions on a distanceplanned
  5. 5Topology on ℝ, integration, and the MVTopen sets; partitions; the Mean Value Theorem via the FTCplanned
  6. 6Taylor series of sin, cos, and expcycling the derivatives at 0 to read off the seriesplanned
  7. 7Expanding a polynomial about a = 4a worked Taylor expansion in (x−a)planned