Maclaurin Series

The central idea

A Maclaurin series is a Taylor series centered at zero: $f(x)=\sum_{n=0}^{\infty}f^{(n)}(0)x^n/n!$ where the series converges to $f$. Standard expansions include $e^x=\sum x^n/n!$, $\sin x=\sum(-1)^nx^{2n+1}/(2n+1)!$, and $\cos x=\sum(-1)^nx^{2n}/(2n)!$.

Definitions, hypotheses, and notation

Known series can be substituted into, multiplied, differentiated, or integrated inside their convergence domains. For example, replacing $x$ by $x^2$ in the exponential series gives $e^{x^2}=\sum x^{2n}/n!$. Such transformations are usually faster and less error-prone than recomputing many derivatives.

Analyticity is stronger than having derivatives of every order. A smooth function can have a Maclaurin series that fails to reproduce it away from zero. Equality requires a remainder tending to zero, so convergence of the coefficient series alone is not the entire justification.

Conceptual meaning

The coefficients encode all derivatives at the origin. A partial sum matches the function's value, slope, curvature, and successively higher derivative data there, turning a transcendental function into a locally accurate polynomial.

A dependable method and decision rule

1. Compute derivatives and evaluate them at zero, or begin from a known base series.
2. Divide the $n$th derivative value by $n!$.
3. Track which powers vanish because of symmetry or derivative cycles.
4. State a radius or interval on which equality is justified.
5. Use a remainder estimate when a numerical accuracy claim is required.

Fully worked example

Graphical or geometric meaning

The constant approximation touches only the value at zero; adding the linear term matches slope; adding the quadratic term matches curvature. Each new term flattens another derivative of the error at the center.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate the polynomial at zero and compare required derivative data.
• Use parity: sine is odd and cosine is even.
• Bound or estimate the first omitted terms.

Coefficients encode derivatives at zero

A Maclaurin expansion is a Taylor expansion centered at zero: the coefficient of $x^n$ is $f^{(n)}(0)/n!$. Equality with the function holds only where the remainder tends to zero. A derivative cycle may reveal the pattern, but substitution into a known base series is often faster; transform the convergence condition along with the variable. Check the constant and first nonzero coefficients directly from derivatives at zero. A finite truncation is a local approximation, and the next nonzero term may predict error size and sign when an appropriate theorem applies. Do not assume every infinitely differentiable function equals its Maclaurin series, because derivative data alone do not guarantee vanishing remainder.

Practice

1. Give the degree-4 Maclaurin polynomial for $\cos x$.
2. Give the first three nonzero terms of $\sin x$.
3. What is the coefficient of $x^3$ in $e^x$?

Connections and next steps

• Taylor Series
• Taylor Polynomial
• Binomial Series