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Sequences & Series · Power & Taylor Series

Maclaurin Series

f(x)=∑n=0∞f(n)(0)n!xn(Taylor series at a=0)f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n \quad \text{(Taylor series at } a = 0\text{)}

A Maclaurin series is a Taylor series centered at a = 0. Common Maclaurin series: eˣ, sin x, cos x, 1/(1-x), ln(1+x).

Worked examples

Write the Maclaurin series for cos x.
  1. f(0) = 1, f'(0) = 0, f''(0) = -1, f'''(0) = 0, f⁽⁴⁾(0) = 1, ...
  2. Even powers only: Σ (-1)ⁿ x^(2n)/(2n)! = 1 - x²/2! + x⁴/4! - ...

Answer: Σ (-1)ⁿ x^(2n)/(2n)!

Write the Maclaurin series for 1/(1-x).
  1. This is a geometric series: Σ xⁿ = 1 + x + x² + x³ + ... for |x| < 1

Answer: Σ xⁿ for |x| < 1

Related formulas

Taylor Series

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