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Calculus I / Infinite Sequences and Series

Derivation — Taylor Expansions for e^x, sin x, and the Binomial Series

This note collects the explicit derivations of the three Maclaurin series that appear most frequently in quantitative finance: exe^x, sin⁡x\sin x (and cos⁡x\cos x), and the binomial series (1+x)k(1+x)^k for real kk. The derivations follow directly from the Taylor coefficient formula, but working through each one carefully builds the muscle memory that makes these series usable: knowing not just the result but how to re-derive it, when it converges, and how to bound the error. Taylor's Inequality is the quantitative tool for the error — it converts "the Taylor series converges to ff" from a qualitative statement into a computable guarantee.
Learning objectives. By the end of this note you should be able to:
  • derive the Maclaurin series for exe^x, sin⁡x\sin x, cos⁡x\cos x, and (1+x)k(1+x)^k from the Taylor formula;
  • state Taylor's Inequality and apply it to show that Tn(x)→f(x)T_n(x) \to f(x);
  • use the Remainder Theorem to determine how many terms are needed for a given accuracy.

In this lesson

  1. Derivation: exe^x
  2. Derivation: sin⁡x\sin x and cos⁡x\cos x
  3. Derivation: The binomial series
  4. Taylor's Inequality in practice

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