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Sequences & Series · Power & Taylor Series

Taylor Series

f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

The Taylor series represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point a.

Worked examples

Find the Taylor series for eˣ centered at a = 0.
  1. f(x) = eˣ. All derivatives are eˣ. f⁽ⁿ⁾(0) = 1 for all n.
  2. Taylor series: Σ xⁿ/n! = 1 + x + x²/2! + x³/3! + ...

Answer: Σ xⁿ/n! (converges for all x)

Find the Taylor series for sin x centered at a = 0.
  1. f(0) = 0, f'(0) = 1, f''(0) = 0, f'''(0) = -1, ...
  2. Only odd powers: Σ (-1)ⁿ x^(2n+1)/(2n+1)! = x - x³/3! + x⁵/5! - ...

Answer: Σ (-1)ⁿ x^(2n+1)/(2n+1)!

Related formulas

Maclaurin SeriesTaylor Polynomial Error Bound

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