What Is a Maclaurin Series?
A Maclaurin series is a way to represent a function as an infinite polynomial centered at \(x = 0\). Instead of working directly with a function such as \(e^x\), \(\sin(x)\), or \(\ln(1+x)\), you build a polynomial from the function’s value and derivatives at zero.
The main idea is that polynomials are often easier to evaluate, graph, and reason about than many elementary functions. A Maclaurin series turns a function into a sum of powers of \(x\):
$$
f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots
$$
A finite part of that series is called a Maclaurin polynomial or partial sum. That partial sum is usually an approximation, not the full function. The more terms you include, the more closely the polynomial may match the function near \(x = 0\), provided the series actually converges at the chosen \(x\) value.
Maclaurin series are a special case of Taylor series. A Taylor series can be centered at any value \(a\); a Maclaurin series is the Taylor series centered at \(a = 0\).
Why Maclaurin Series Matter
Maclaurin series matter because they let you replace complicated functions with polynomials. That is useful in calculus, numerical methods, physics, engineering, and computer-based approximation.
For example, a calculator, graphing tool, or numerical method may not work with \(\sin(x)\) directly in every step. It may instead use a polynomial approximation that is accurate enough over a certain range. Maclaurin polynomials also help students see why functions behave the way they do near zero: the first few terms often reveal the function’s local shape, slope, curvature, and symmetry.
The important caution is that a Maclaurin polynomial is local. It is built around zero, so it is usually most reliable near zero. Some functions have series that converge for every real \(x\), while others converge only inside a limited interval.
Key Terms to Know
-
Taylor series: A power series built from a function’s derivatives at a chosen center \(a\).
-
Maclaurin series: A Taylor series centered at \(0\).
-
Maclaurin polynomial: A finite partial sum of a Maclaurin series.
-
Term: One addend in the series, such as \(\frac{x^3}{3!}\) or \(-\frac{x^5}{5!}\).
-
Partial sum: The sum of a limited number of terms.
-
Radius of convergence: The distance from the center where the power series is guaranteed to converge inside that radius.
-
Interval of convergence: The set of \(x\) values where the series converges.
-
Absolute error: The size of the difference between an exact or reference value and an approximation.
-
Relative error: The error compared with the size of the exact or reference value, often written as a percentage.
-
Factorial: The product \(n! = n \times (n-1) \times \cdots \times 2 \times 1\), with \(0! = 1\).
How Maclaurin Series Work
For a sufficiently smooth function, the Maclaurin polynomial uses the function’s derivatives at zero:
$$
P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!}x^k
$$
Where:
-
\(P_n(x)\) is the Maclaurin polynomial through degree \(n\).
-
\(f^{(k)}(0)\) is the \(k\)th derivative of the function evaluated at \(0\).
-
\(k!\) is the factorial of \(k\).
-
\(x\) is the value where you want to approximate the function.
The infinite Maclaurin series is:
$$
f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f^{(3)}(0)}{3!}x^3 + \cdots
$$
A partial sum stops after a finite number of terms. Its polynomial subscript is the degree, while this calculator's term controls count generated nonzero terms. For example, the first three sine terms form the degree-five polynomial \(P_5(x)\), also described as a 3-term partial sum:
$$
f(x) \approx P_n(x)
$$
This approximation can be very accurate near \(x = 0\), but accuracy depends on the function, the number of terms used, the selected \(x\) value, and whether the series converges there.
Common Supported Maclaurin Series
The calculator works with a built-in set of common Maclaurin series. These are the main forms behind the supported functions:
| Function |
Maclaurin series |
Typical convergence interval |
| \(\sin(x)\) |
\(\sum_{i=0}^{\infty} (-1)^i\frac{x^{2i+1}}{(2i+1)!}\) |
All real \(x\) |
| \(\cos(x)\) |
\(\sum_{i=0}^{\infty} (-1)^i\frac{x^{2i}}{(2i)!}\) |
All real \(x\) |
| \(e^x\) |
\(\sum_{i=0}^{\infty} \frac{x^i}{i!}\) |
All real \(x\) |
| \(\frac{1}{1-x}\) |
\(\sum_{i=0}^{\infty} x^i\) |
\(\lvert x \rvert < 1\) |
| \(\frac{1}{1+x}\) |
\(\sum_{i=0}^{\infty} (-x)^i\) |
\(\lvert x \rvert < 1\) |
| \(\ln(1+x)\) |
\(\sum_{i=0}^{\infty} (-1)^i\frac{x^{i+1}}{i+1}\) |
\(-1 < x \le 1\) |
| \(\sqrt{1+x}\) |
\(\sum_{i=0}^{\infty} \binom{1/2}{i}x^i\) |
Usually considered on \(-1 \le x \le 1\) for real values |
| \(\arctan(x)\) |
\(\sum_{i=0}^{\infty} (-1)^i\frac{x^{2i+1}}{2i+1}\) |
\(-1 \le x \le 1\) |
| \(\tan(x)\) |
\(x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots\) |
\(\lvert x \rvert < \frac{\pi}{2}\) |
The convergence interval is just as important as the formula. A partial sum can still produce a number outside the interval of convergence, but that number should not be treated as a reliable approximation of the function.
Endpoint inclusion is handled exactly as shown in the table. A value even slightly outside an included endpoint is outside the convergence interval.
Examples of Maclaurin Series in Practice
Example 1: Approximating \(\sin(0.5)\)
The Maclaurin series for \(\sin(x)\) begins:
$$
\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
$$
Using the first three nonzero terms at \(x = 0.5\):
$$
P(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!}
$$
Substitute \(x = 0.5\):
$$
P(0.5) = 0.5 - \frac{0.5^3}{6} + \frac{0.5^5}{120}
$$
$$
P(0.5) \approx 0.47942708
$$
The direct value of \(\sin(0.5)\) is approximately \(0.47942554\), so the absolute error is about:
$$
\lvert 0.47942554 - 0.47942708 \rvert \approx 0.00000154
$$
This is a strong approximation because \(0.5\) is close to zero and the sine series converges for every real \(x\).
Example 2: Approximating \(e^{0.5}\)
The Maclaurin series for \(e^x\) is:
$$
e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots
$$
Using the first four terms:
$$
P(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}
$$
At \(x = 0.5\):
$$
P(0.5) = 1 + 0.5 + \frac{0.5^2}{2} + \frac{0.5^3}{6}
$$
$$
P(0.5) \approx 1.64583333
$$
The direct value of \(e^{0.5}\) is approximately \(1.64872127\), so the absolute error is about \(0.00288794\). Adding more terms generally improves this approximation because the exponential series converges for all real \(x\).
Example 3: A Convergence Edge Case
The geometric Maclaurin series for \(\frac{1}{1-x}\) is:
$$
\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots
$$
This series converges only when \(\lvert x \rvert < 1\).
At \(x = 0.5\), the first four terms give:
$$
1 + 0.5 + 0.5^2 + 0.5^3 = 1.875
$$
The exact value is:
$$
\frac{1}{1-0.5} = 2
$$
The approximation is reasonable and will improve as more terms are added.
At \(x = 1.2\), however, the same series is outside its convergence interval. A finite partial sum can still be calculated, but the infinite series does not converge to \(\frac{1}{1-x}\) there. This is why the radius or convergence status should be checked before trusting a result.
How to Interpret the Result
A Maclaurin series result should be read as a comparison between a direct function value and a partial polynomial approximation.
The main result values mean:
-
Detected function: The supported function form recognized from the expression input.
-
Exact value: The direct reference value of the function at the selected \(x\).
-
Series approximation: The Maclaurin partial sum using the terms currently included in the displayed polynomial.
-
Absolute error: The size of the difference between the exact value and the approximation.
-
Relative error: The absolute error divided by the size of the exact value, shown as a percentage when that comparison is meaningful.
-
Convergence status: Whether the selected \(x\) is within the convergence range used for the selected series.
-
Radius note: A reminder of the function’s convergence radius or interval behavior.
-
Term list and partial sums: A step-by-step view of how each additional term changes the approximation.
-
Graph: A visual comparison between the original function and the Maclaurin polynomial over the selected range.
The approximation is based on the revealed nonzero terms, not always the total number generated. The formula uses \(P_d(x)\), where \(d\) is the exponent of its last revealed term. For example, three revealed sine terms are \(P_5(x)\), a 3-term partial sum.
Absolute error is calculated as:
$$
\text{absolute error} = \lvert \text{exact value} - \text{series approximation} \rvert
$$
Relative error is calculated as:
$$
\text{relative error} =
\frac{\lvert \text{exact value} - \text{series approximation} \rvert}
{\lvert \text{exact value} \rvert}
\times 100\%
$$
A small absolute error means the approximation is numerically close to the exact value. A small relative error means the error is small compared with the size of the exact value. When the exact value is zero or extremely close to zero, relative error may be unavailable or misleading because it requires division by a value near zero.
Common Mistakes and Misconceptions
One common mistake is assuming that more terms always solve every problem. More terms usually help when the series converges and the input is reasonably close to the center, but they do not fix a point outside the convergence interval.
Another mistake is confusing a Maclaurin series with any Taylor series. Every Maclaurin series is a Taylor series, but not every Taylor series is a Maclaurin series. Maclaurin series are always centered at \(0\).
Users may also assume that the total term count is always the number of terms used in the displayed approximation. In this calculator, the displayed approximation follows the currently revealed nonzero terms. This count is distinct from polynomial degree for sparse series such as sine and cosine. Use the reveal controls or show all terms when you want the full generated partial sum.
A related mistake is typing unsupported expressions. The calculator is designed for the listed supported functions, not arbitrary symbolic expressions, shifted functions, or custom combinations of functions.
Numeric input format is another frequent source of confusion. Decimal and scientific notation are appropriate for numeric fields, but symbolic values such as \(\pi/2\), \(\pi\), \(e\), or fractions such as \(1/2\) should be entered as decimal approximations when a numeric field is required.
Finally, do not read a missing relative error as zero error. A dash or unavailable relative error usually means the exact value is too close to zero, non-finite, or otherwise not suitable for a percentage comparison.
When to Use Maclaurin Series
Use Maclaurin series when you want to:
-
Approximate a supported function near \(x = 0\).
-
See how a polynomial approximation improves as terms are added.
-
Compare an approximation with a direct function value.
-
Study convergence, partial sums, and approximation error.
-
Build intuition for calculus topics such as Taylor polynomials, power series, and numerical approximation.
-
Visualize how a polynomial can match a function near the center but drift away elsewhere.
Maclaurin series are especially helpful for learning. They show not only the final approximation, but also how each term contributes to the result.
Limitations and Things to Keep in Mind
Maclaurin series are centered at zero. If you need an approximation around another point, a Taylor series centered at that point may be more appropriate.
The calculator supports only these function families and listed aliases: \(\sin(x)\), \(\cos(x)\), \(e^x\), \(\frac{1}{1-x}\), \(\frac{1}{1+x}\), \(\ln(1+x)\), \(\sqrt{1+x}\), \(\arctan(x)\), and \(\tan(x)\). It does not symbolically derive Maclaurin series for arbitrary functions.
The calculations are real-valued. Complex inputs and complex-valued outputs are not supported.
The number of generated terms is limited, and displayed values are rounded. Main numeric outputs are shown to a practical number of decimal places, so very small differences may appear as zero. Rounding is useful for readability, but it should not be confused with a proof of exact equality.
The convergence warning is essential. For finite-radius series, a displayed partial sum outside the convergence interval may be mathematically divergent or misleading. A number can be displayed even when the underlying infinite series does not converge to the function.
The exact comparison is a pointwise comparison at the selected \(x\). It does not prove that the approximation is good across the whole graph range. The graph is a visual aid, not a rigorous error bound.
The graph deliberately breaks the exact-function curve at known poles of the reciprocal and tangent functions. This prevents a line segment from being drawn across a vertical asymptote; the polynomial approximation remains continuous.
Some inputs make the original function undefined or non-finite. Examples include \(\frac{1}{1-x}\) at \(x = 1\), \(\frac{1}{1+x}\) at \(x = -1\), \(\ln(1+x)\) for \(x \le -1\), \(\sqrt{1+x}\) for \(x < -1\), and \(\tan(x)\) near values where \(\cos(x) = 0\).
For homework, teaching, or exploratory work, the displayed exact-vs-approximation error is often enough to understand the result. For engineering, safety-critical, or formal numerical work, use a rigorous error bound or a validated numerical method.
How to Use This Calculator
-
Enter a supported function form, such as
sin(x), cos(x), exp(x), e^x, 1/(1-x), 1/(1+x), ln(1+x), sqrt(1+x), atan(x), or tan(x).
-
Enter the \(x\) value where you want to evaluate the function and its Maclaurin approximation.
-
Choose a whole-number total of generated nonzero terms from 1 through 30.
-
Reveal nonzero terms step by step or show all generated terms. The displayed approximation uses the revealed-term count, while its \(P_d\) label uses the resulting polynomial degree.
-
Set the graph range with
x-min and x-max. These values must be different.
-
Choose a whole-number graph sample count from 40 through 240 if you want to change the plotted sampling density.
-
Review the detected function, exact value, series approximation, absolute error, relative error, convergence status, radius note, term list, and graph.
-
Check the convergence message before relying on the approximation.
Frequently Asked Questions
What is the difference between a Taylor series and a Maclaurin series?
A Taylor series is centered at a chosen value \(a\). A Maclaurin series is the special case where the center is \(a = 0\). Because this calculator uses series centered at zero, it is working with Maclaurin series rather than Taylor series centered at other points.
Why does the approximation usually work best near zero?
The polynomial is built from information about the function at \(x = 0\). Near zero, those derivative-based terms often describe the function well. Farther from zero, the polynomial may need many more terms, and for finite-radius series it may stop converging altogether.
Does adding more terms always make the result better?
Not always. Adding terms usually improves the approximation inside the convergence interval, especially near zero. Outside the convergence interval, the partial sums may diverge or behave unpredictably, so more terms can make the displayed approximation worse rather than better.
Why is the relative error sometimes unavailable?
Relative error compares the absolute error with the size of the exact value. If the exact value is zero, extremely close to zero, undefined, or non-finite, that percentage comparison is not meaningful. In that case, the absolute error and the function’s domain are usually more useful.
Can I enter pi/2, π, e, or fractions as numeric values?
Numeric fields are intended for finite decimal or scientific-notation values. Enter a decimal approximation such as 1.5708 instead of pi/2, or 0.5 instead of 1/2.
What does “outside radius” mean?
It means the chosen \(x\) value is outside the convergence range used for that function’s Maclaurin series. The calculator may still show a finite partial sum, but the infinite series is not expected to converge to the function at that point. Treat outside-radius results as exploratory, not reliable approximations.
Why does the graph sometimes look good near zero but poor farther away?
A Maclaurin polynomial is centered at zero. It often matches the function closely near zero and then drifts away as \(x\) moves farther from the center. This effect is especially visible for finite partial sums and functions with limited convergence intervals.
Sources and References
Books
-
Gilbert Strang and Edwin “Jed” Herman. Calculus Volume 2. OpenStax, 2016. Sections 3.6, 6.1, 6.3, and 6.4. https://openstax.org/books/calculus-volume-2/pages/1-introduction
-
Matthew Boelkins, David Austin, Christina Safranski, and Steven Schlicker. Active Calculus: Single Variable, 2nd ed. Grand Valley State University Libraries, 2025. Section 8.4, “Taylor Series.” https://activecalculus.org/single-alt/sec-8a-4-taylor-series.html
Online and Official Sources
-
National Institute of Standards and Technology. “DLMF §4.19 Maclaurin Series and Laurent Series.” NIST Digital Library of Mathematical Functions, Version 1.2.7, release date June 15, 2026. https://dlmf.nist.gov/4.19
-
National Institute of Standards and Technology. “DLMF §24.15 Related Sequences of Numbers.” NIST Digital Library of Mathematical Functions, Version 1.2.7, release date June 15, 2026. https://dlmf.nist.gov/24.15
-
MIT OpenCourseWare. “Session 98: Taylor’s Series.” 18.01SC Single Variable Calculus, Fall 2010. https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/pages/unit-5-exploring-the-infinite/part-b-taylor-series/session-98-taylors-series/
