Maclaurin Series — Definition, Formula & Examples

Key Formula

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}\, x^n = f(0) + f'(0)\,x + \frac{f''(0)}{2!}\,x^2 + \frac{f'''(0)}{3!}\,x^3 + \cdots

Where:

$f(x)$ = The function being expanded as a power series
$f^{(n)}(0)$ = The nth derivative of f evaluated at x = 0
$n!$ = n factorial (the product 1 × 2 × 3 × … × n)
$x$ = The variable; the series is in powers of x
$n$ = The index of summation, starting from 0

Worked Example

Problem: Find the Maclaurin series for f(x) = e^x.

Step 1: Compute the derivatives of f(x) = e^x. Every derivative of e^x is e^x.

f(x) = e^x,\quad f'(x) = e^x,\quad f''(x) = e^x,\quad f'''(x) = e^x, \ldots

Step 2: Evaluate each derivative at x = 0. Since e^0 = 1, every derivative at zero equals 1.

f(0) = 1,\quad f'(0) = 1,\quad f''(0) = 1,\quad f'''(0) = 1, \ldots

Step 3: Substitute into the Maclaurin series formula, replacing f^(n)(0) with 1 for every n.

e^x = \sum_{n=0}^{\infty} \frac{1}{n!}\,x^n = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

Step 4: State the interval of convergence. This series converges for all real numbers x.

\text{Interval of convergence: } (-\infty,\,\infty)

Answer: The Maclaurin series for e^x is

\sum_{n=0}^{∞} \dfrac{x^n}{n!} = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6} + \ldots

, converging for all x.

Another Example

This example shows a function whose Maclaurin series has only even powers of x (all odd-power coefficients are zero), illustrating that not every term in the general formula necessarily appears.

Problem: Find the Maclaurin series for f(x) = cos(x) through the x⁶ term.

Step 1: Compute successive derivatives of cos(x). The derivatives cycle with period 4.

f(x)=\cos x,\; f'(x)=-\sin x,\; f''(x)=-\cos x,\; f'''(x)=\sin x,\; f^{(4)}(x)=\cos x, \ldots

Step 2: Evaluate at x = 0. Recall sin(0) = 0 and cos(0) = 1.

f(0)=1,\; f'(0)=0,\; f''(0)=-1,\; f'''(0)=0,\; f^{(4)}(0)=1,\; f^{(5)}(0)=0,\; f^{(6)}(0)=-1

Step 3: Notice the pattern: odd-order derivatives vanish at 0, so all odd powers of x drop out. Only even powers survive.

\cos x = 1 + 0\cdot x + \frac{-1}{2!}x^2 + 0 + \frac{1}{4!}x^4 + 0 + \frac{-1}{6!}x^6 + \cdots

Step 4: Simplify and write in sigma notation.

\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}\,x^{2n} = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots

Answer: The Maclaurin series for cos(x) is

\sum_{n=0}^{∞} \dfrac{(-1)^n}{(2n)!}\,x^{2n} = 1 - \dfrac{x^2}{2} + \dfrac{x^4}{24} - \dfrac{x^6}{720} + \ldots

, converging for all x.

Frequently Asked Questions

What is the difference between a Maclaurin series and a Taylor series?

A Maclaurin series is simply a Taylor series centered at a = 0. A Taylor series can be centered at any point a, using powers of (x − a) and derivatives evaluated at a. When you set a = 0 in the Taylor series formula, you get the Maclaurin series.

When do you use a Maclaurin series?

You use a Maclaurin series when you need to approximate a function near x = 0, evaluate difficult integrals or limits, or represent a function as a polynomial for computation. It is especially useful in physics and engineering when x is small, because just a few terms can give a very accurate approximation.

Does a Maclaurin series always converge to the original function?

Not always. The series must first converge (which happens within the radius of convergence), and even if it converges, it might not equal f(x) at every point. A classic counterexample is f(x) = e^(−1/x²) (defined as 0 at x = 0), whose Maclaurin series is identically zero yet the function is not zero for x ≠ 0. For most standard functions like e^x, sin x, and cos x, the series does converge to f(x) for all x.

Maclaurin Series vs. Taylor Series

	Maclaurin Series	Taylor Series
Center point	Always centered at a = 0	Centered at any chosen point a
Formula	$\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}\,x^n$	$\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$
Powers of	Powers of x	Powers of (x − a)
Best approximation near	x = 0	x = a (your chosen center)
When to use	When the function behavior near 0 is of interest	When you need accuracy near a point a ≠ 0

Why It Matters

Maclaurin series appear frequently in AP Calculus BC and university calculus courses, where you are expected to derive series for standard functions and use them to compute limits, integrals, and approximations. In physics, truncating the series after a few terms yields polynomial approximations (like sin x ≈ x for small x) that simplify complex problems. Knowing the common Maclaurin series for e^x, sin x, cos x, 1/(1 − x), and ln(1 + x) by heart saves significant time on exams and in applications.

Common Mistakes

Mistake: Forgetting to divide by n! in each term.

Correction: Every coefficient in the Maclaurin series includes a factor of 1/n!. For instance, the x² term for e^x is x²/2!, not just x². Always write out the factorial denominator explicitly until the pattern is clear.

Mistake: Using the wrong derivative values by evaluating at a point other than 0.

Correction: A Maclaurin series requires all derivatives evaluated at x = 0, not at some other point. If you evaluate derivatives at x = a (where a ≠ 0), you are building a Taylor series centered at a, which is a different expansion.

Related Terms

Taylor Series — General form; Maclaurin is the a = 0 case
Power Series — A Maclaurin series is a specific power series
Power Series Convergence — Determines where the series converges
Convergence Tests — Used to find the radius of convergence
Factorial — n! appears in every Maclaurin series denominator
Sigma Notation — Compact notation for writing the series
Function — The object being represented as a series