Maclaurin Series Calculator
Free Maclaurin series Calculator for sequences. Enter values to get step-by-step solutions with formulas and graphs. Get results you can export or share.
Calculator
Adjust values & calculateTerm-by-Term Expansion
Formula
The Maclaurin series expands f(x) as an infinite sum of terms involving the nth derivative of f evaluated at 0, multiplied by x^n/n!. It is a Taylor series centered at a = 0. The series converges to f(x) within the radius of convergence R.
Last reviewed: December 2025
Worked Examples
Example 1: Approximating e^1 with 8 Terms
Example 2: Computing sin(0.5) via Maclaurin Series
Background & Theory
The Maclaurin Series Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Maclaurin Series Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
f(x) = sum of f^(n)(0) * x^n / n!
The Maclaurin series expands f(x) as an infinite sum of terms involving the nth derivative of f evaluated at 0, multiplied by x^n/n!. It is a Taylor series centered at a = 0. The series converges to f(x) within the radius of convergence R.
Worked Examples
Example 1: Approximating e^1 with 8 Terms
Problem: Calculate e using the first 8 terms of the Maclaurin series for e^x at x = 1.
Solution: e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7!\nAt x = 1: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + 1/5040\n= 1 + 1 + 0.5 + 0.16667 + 0.04167 + 0.00833 + 0.00139 + 0.000198\n= 2.71825\nExact e = 2.71828...\nError = 0.00003 (5 digits of accuracy with 8 terms)
Result: 8-term approximation = 2.71825 | Exact = 2.71828 | Error = 3e-5
Example 2: Computing sin(0.5) via Maclaurin Series
Problem: Approximate sin(0.5) using the Maclaurin series with 5 non-zero terms.
Solution: sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9!\nsin(0.5) = 0.5 - 0.125/6 + 0.03125/120 - 0.0078125/5040 + 0.001953125/362880\n= 0.5 - 0.020833 + 0.000260 - 0.00000155 + 0.0000000054\n= 0.479426\nExact sin(0.5) = 0.479426...\nThe series converges extremely fast for |x| < 1
Result: sin(0.5) = 0.479426 | Full precision reached with just 4 terms
Frequently Asked Questions
What is a Maclaurin series and how does it differ from a Taylor series?
A Maclaurin series is a special case of the Taylor series centered at x = 0. The general Taylor series of f(x) centered at x = a is the sum of f^(n)(a) * (x-a)^n / n! for n = 0, 1, 2, and so on. When a = 0, this becomes the Maclaurin series: sum of f^(n)(0) * x^n / n!. Named after Scottish mathematician Colin Maclaurin, this series represents a function as an infinite polynomial around the origin. The Maclaurin series is particularly useful because many common functions have clean, memorable series when expanded at zero. Both Taylor and Maclaurin series converge to the function within the radius of convergence, providing polynomial approximations of arbitrary precision.
What are the most important Maclaurin series to memorize?
The essential Maclaurin series include: e^x = 1 + x + x^2/2! + x^3/3! + ... (converges for all x). sin(x) = x - x^3/3! + x^5/5! - ... (all x). cos(x) = 1 - x^2/2! + x^4/4! - ... (all x). ln(1+x) = x - x^2/2 + x^3/3 - ... (for -1 < x <= 1). 1/(1-x) = 1 + x + x^2 + x^3 + ... (for |x| < 1). arctan(x) = x - x^3/3 + x^5/5 - ... (for |x| <= 1). (1+x)^k = 1 + kx + k(k-1)x^2/2! + ... (the binomial series, for |x| < 1). These series are building blocks for deriving more complex expansions through substitution, differentiation, and multiplication.
How do you determine the radius of convergence of a Maclaurin series?
The radius of convergence R determines the interval (-R, R) where the series converges. The ratio test is the most common method: R = lim |a_n / a_(n+1)| as n approaches infinity, where a_n is the coefficient of x^n. Alternatively, the root test gives 1/R = lim |a_n|^(1/n). For e^x, R = lim (n+1)!/n! = lim (n+1) = infinity, so it converges everywhere. For ln(1+x), R = lim n/(n+1) = 1, so it converges for |x| < 1. The series may or may not converge at the endpoints x = R and x = -R, which must be checked separately. Functions with singularities in the complex plane have R equal to the distance from the center to the nearest singularity.
How do you derive a new Maclaurin series from known ones?
Several techniques generate new series from known ones. Substitution: replacing x with g(x) in a known series gives the series for f(g(x)). For example, e^(-x^2) = 1 - x^2 + x^4/2! - x^6/3! + ... by substituting -x^2 into the e^x series. Differentiation: differentiating term by term gives the series for f'(x). Since 1/(1-x) = 1 + x + x^2 + ..., differentiating gives 1/(1-x)^2 = 1 + 2x + 3x^2 + ... Integration: integrating 1/(1+x) = 1 - x + x^2 - ... gives ln(1+x) = x - x^2/2 + x^3/3 - ... Multiplication: multiplying two series gives the series for the product function. These techniques avoid computing derivatives directly.
How accurate is a partial sum of a Maclaurin series?
The accuracy depends on three factors: the number of terms used, the value of x, and the function being approximated. Near x = 0, even a few terms provide excellent accuracy. For e^1, 10 terms give 7 digits of accuracy. For sin(0.1), just 3 terms give 12 digits of accuracy. However, accuracy degrades as |x| increases. For e^10, you need about 40 terms for 10-digit accuracy. The error of an n-term partial sum is bounded by the Lagrange error bound: |error| <= M * |x|^(n+1) / (n+1)! where M bounds the (n+1)th derivative. For alternating series, the error is bounded by the absolute value of the first omitted term, which is often a tighter bound.
What is the relationship between Maclaurin series and Euler's formula?
Euler's formula e^(ix) = cos(x) + i*sin(x) connects the exponential, sine, and cosine Maclaurin series in a profound way. Substituting ix into the e^x series: e^(ix) = 1 + ix - x^2/2! - ix^3/3! + x^4/4! + ix^5/5! - ... Separating real and imaginary parts: the real terms give 1 - x^2/2! + x^4/4! - ... = cos(x), and the imaginary terms give x - x^3/3! + x^5/5! - ... = sin(x). The special case x = pi gives Euler's identity e^(i*pi) + 1 = 0, often called the most beautiful equation in mathematics. This relationship unifies exponential and trigonometric functions through complex numbers and their Maclaurin series.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy