A Maclaurin series is a Taylor series centered at zero — the most useful special case, where a function is written as a clean sum of powers of $x$ built from its derivatives at the origin.
What is a Maclaurin series?
A Maclaurin series expands a function as an infinite sum of terms $c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots$, where each coefficient comes from a derivative of the function evaluated at $x=0$. Because the center is zero, the awkward $(x-a)$ factors of a general Taylor series collapse to simple powers of $x$ — which is exactly why Maclaurin series are the version you meet first and use most.
The Maclaurin series formula
Setting $a=0$ in the Taylor formula gives the Maclaurin series:
$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\,x^n = f(0) + f'(0)\,x + \frac{f”(0)}{2!}x^2 + \frac{f”'(0)}{3!}x^3 + \cdots$$Each coefficient is one derivative of $f$ at zero, divided by a factorial. That single simplification — evaluating everything at $0$ — is what makes the standard series so tidy.
Why center at zero?
- Cleaner algebra: powers of $x$ instead of $(x-a)^n$.
- Easy derivatives: for many functions, $f^{(n)}(0)$ follows a simple repeating pattern.
- Best near the origin: the approximation is sharpest around $x=0$, which is where many physics and ML quantities live (small angles, small weight updates).
How to find a Maclaurin series, step by step
- Differentiate the function several times.
- Evaluate each derivative at $x=0$ to get $f(0), f'(0), f”(0), \dots$
- Divide by $n!$ to form the coefficient of $x^n$.
- Write the sum $f(0)+f'(0)x+\frac{f”(0)}{2!}x^2+\cdots$ to your chosen order.
Or let the Taylor & Maclaurin series calculator do it — set the center to $0$ and it returns the Maclaurin series with a graph.
Worked example: the Maclaurin series of cos x
The derivatives of $\cos x$ cycle $\cos x \to -\sin x \to -\cos x \to \sin x \to \cos x$. Evaluated at $0$ they give $1, 0, -1, 0, 1, \dots$, so only even powers appear:
| n | $f^{(n)}(x)$ | $f^{(n)}(0)$ | term |
|---|---|---|---|
| 0 | $\cos x$ | 1 | $1$ |
| 1 | $-\sin x$ | 0 | $0$ |
| 2 | $-\cos x$ | $-1$ | $-x^2/2!$ |
| 3 | $\sin x$ | 0 | $0$ |
| 4 | $\cos x$ | 1 | $x^4/4!$ |
So $\cos x \approx 1 – \dfrac{x^2}{2!} + \dfrac{x^4}{4!} – \dfrac{x^6}{6!} + \cdots$
The essential Maclaurin series
These show up again and again — worth committing to memory:
| Function | Maclaurin series | Converges for |
|---|---|---|
| $e^x$ | $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ | all $x$ |
| $\sin x$ | $x – \frac{x^3}{3!} + \frac{x^5}{5!} – \cdots$ | all $x$ |
| $\cos x$ | $1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \cdots$ | all $x$ |
| $\frac{1}{1-x}$ | $1 + x + x^2 + x^3 + \cdots$ | $|x|<1$ |
| $\ln(1+x)$ | $x – \frac{x^2}{2} + \frac{x^3}{3} – \cdots$ | $-1 < x \le 1$ |
| $\arctan x$ | $x – \frac{x^3}{3} + \frac{x^5}{5} – \cdots$ | $|x|\le 1$ |
| $(1+x)^p$ | $1 + px + \frac{p(p-1)}{2!}x^2 + \cdots$ | $|x|<1$ |
A quick derivation: the geometric series
The simplest Maclaurin series of all is $\frac{1}{1-x}$. Its derivatives at $0$ are $1, 1, 2, 6, 24, \dots = n!$, so each coefficient $\frac{n!}{n!}=1$, giving:
$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots \quad (|x|<1)$$This is the geometric series — and many other Maclaurin series (like $\ln(1+x)$ and $\arctan x$) are derived by integrating or substituting into it.
Where Maclaurin series show up
Because they are centered at the origin, Maclaurin series are the natural tool whenever a quantity is small:
- Physics: $\sin\theta \approx \theta$ and $\cos\theta \approx 1-\frac{\theta^2}{2}$ for small angles.
- Numerical computing: evaluating $e^x$ or $\sin x$ from their series.
- Machine learning: activation functions and loss surfaces are routinely analyzed through their Maclaurin (low-order Taylor) expansions near a point.
🤖 ML insight
The softmax and sigmoid functions are often approximated by the first terms of their Maclaurin series to reason about gradients near zero. The same low-order expansion underlies why small weight updates behave predictably during training. Need the derivatives? Use our derivative calculator.
Frequently asked questions
What is a Maclaurin series?
What is the Maclaurin series formula?
Is a Maclaurin series the same as a Taylor series?
What is the Maclaurin series of cos x?
Why do some Maclaurin series only converge for |x| < 1?
Key takeaways
A Maclaurin series is the centered-at-zero Taylor series, giving clean powers of $x$ and a short list of must-know expansions. Put it to work with the Maclaurin series calculator (set the center to 0), read the broader Taylor series guide, or check the formal reference on Wikipedia.