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Maclaurin series

A Maclaurin expansion uses derivatives at zero to approximate a function near zero.

The coefficients are f(0), f′(0), f″(0)/2!, and so on. A polynomial truncated after finitely many terms is an approximation, even when the full infinite series equals the function on a convergence interval.

State the requested order and any validity condition. Substituting a value outside a known convergence interval does not become justified by adding more terms. If the function is not defined at zero, its Maclaurin expansion cannot be obtained by this direct derivative formula.

Worked example

Give eˣ through the x³ term.

  1. Every derivative of eˣ is eˣ.
  2. Every derivative at zero equals 1.
  3. Divide each coefficient by the appropriate factorial.

Answer: 1 + x + x²/2 + x³/6 + O(x⁴)

Revise first: Differentiating powers.

A finite expansion still has a domain

A Maclaurin polynomial matches derivatives at zero. Its xⁿ coefficient is f^(n)(0)/n!, not just the nth derivative. A truncated expansion is an approximation unless the original function is a polynomial of that degree.

The geometric series 1/(1 − u) = 1 + u + u² + ⋯ converges only for |u| < 1. Integrating or differentiating it gives useful expansions, while preserving a region of convergence. Substituting u = ax changes that region to |ax| < 1.

ln(1 + 2x) through x³

  1. Differentiate: 2/(1 + 2x) = 2 − 4x + 8x² − ⋯ for |2x| < 1.
  2. Integrate term by term: ln(1 + 2x) = 2x − 2x² + (8/3)x³ − ⋯.
  3. The constant is zero because ln(1) = 0.
  4. The expansion converges for |x| < 1/2. A third-degree truncation should not be used far from zero without an error bound.
What happens to the convergence interval if 2x is replaced by −3x?

It becomes |x| < 1/3. The signs of the coefficients change as well.

Take the idea further

Differentiate the geometric series to expand (1 − ax)^−2. Its xⁿ coefficient is (n + 1)aⁿ.

More advanced methods

Practise maclaurin series

Course mapping

These specification references show where the topic occurs. The questions cover only some parts of each topic.

Next practice: Geometric series.