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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).

  • 2 worked examples
  • 18 in Sequences & Series

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)!

1 more worked examplePremium

Write the Maclaurin series for 1/(1-x).

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Taylor Series

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Common questions

What is the Maclaurin Series?

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). It is one of the sequences & series formulas in the CalcRef reference.

How do you use the Maclaurin Series?

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

Practice

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