Taylor & Maclaurin Series Calculator

Series Expansion
Click buttons above or type directly. Use ^ for powers, e.g. x^2

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Series Expansion

ฮฃ

Enter a function

Calculate Taylor or Maclaurin series expansion with step-by-step solutions.

Convergence Graph

Calculate a series to see the function vs approximation graph.

Terms: 5

Python Compiler

What is a Taylor Series?

A Taylor series represents a function as an infinite sum of polynomial terms calculated from the function's derivatives at a single point. It is one of the most powerful tools in calculus โ€” enabling us to approximate complex functions like sin(x), ex, and ln(x) using simple polynomials.

f(x)
Function
=
ฮฃ
f(n)(a)
nth Derivative
/
n!
Factorial
ยท
(xโˆ’a)n
Power Term
๐Ÿ’ก
Why does it work? Each term matches one more derivative of the original function at the center point. With enough terms, the polynomial approximation becomes indistinguishable from the original function โ€” at least within the radius of convergence.

Common Maclaurin Series

These series are used so frequently in mathematics and physics that they are worth memorizing.

โ— Exponential

ex = 1 + x + xยฒ/2! + xยณ/3! + โ€ฆ

R = โˆž

โ— Sine

sin(x) = x โˆ’ xยณ/3! + x5/5! โˆ’ โ€ฆ

R = โˆž

โ— Cosine

cos(x) = 1 โˆ’ xยฒ/2! + x4/4! โˆ’ โ€ฆ

R = โˆž

โ— Natural Log

ln(1+x) = x โˆ’ xยฒ/2 + xยณ/3 โˆ’ โ€ฆ

R = 1

โ— Geometric

1/(1โˆ’x) = 1 + x + xยฒ + xยณ + โ€ฆ

R = 1

โ— Square Root

โˆš(1+x) = 1 + x/2 โˆ’ xยฒ/8 + โ€ฆ

R = 1

Understanding Convergence

Not every Taylor series converges everywhere. The radius of convergence R tells you how far from the center point the series reliably approximates the function.

โ— R = โˆž

Functions like ex, sin(x), and cos(x) converge for all real x. Their series approximation works everywhere.

โ— Finite R

Functions like ln(1+x) and 1/(1โˆ’x) only converge within a limited interval around the center. Beyond that, the series diverges.

โ— Singularities

The radius of convergence equals the distance to the nearest singularity (point where the function is undefined), even in the complex plane.

๐Ÿ‘‰
Try it! Enter ln(1+x) and increase terms to 15. Watch how the graph matches well for |x| < 1 but diverges wildly beyond x = 1.

Real-World Applications

Taylor series arenโ€™t just theoretical โ€” they power real technology and science every day.

Calculator Chips

Your calculator computes sin(x) and cos(x) using polynomial approximations derived from Taylor series. Hardware implements these as fast multiplications and additions.

Physics Approximations

sin(ฮธ) โ‰ˆ ฮธ for small angles simplifies pendulum equations. Many physics formulas are first-order Taylor approximations.

Machine Learning

Gradient descent uses first-order Taylor approximation. Newtonโ€™s method uses second-order. Higher-order optimization uses more terms.

Frequently asked

A Taylor series expands a function f(x) around any point a using f(x) = ฮฃ f(n)(a)/n! ยท (xโˆ’a)n. A Maclaurin series is the special case where a = 0, so f(x) = ฮฃ f(n)(0)/n! ยท xn. Both represent functions as infinite polynomial sums โ€” this calculator supports both.
Depends on the function and how far from the center you evaluate. Near the center, 5โ€“7 terms often give excellent accuracy. Farther away โ€” or for functions with small convergence radii โ€” you may need 15โ€“20 terms. Use the interactive graph with the term slider to watch convergence in real time.
The radius of convergence R is the distance from the center within which the series converges to the actual function. For |xโˆ’a| < R, more terms get closer to the true value; for |xโˆ’a| > R the series diverges. Common values: ex has R = โˆž, ln(1+x) has R = 1, tan(x) has R = ฯ€/2.
Replace the integrand with its Taylor polynomial, then integrate term-by-term. For example, โˆซ01 eโˆ’xยฒ dx โ‰ˆ 1 โˆ’ 1/3 + 1/10 โˆ’ โ€ฆ โ‰ˆ 0.7468. Use Integral Approx mode for automatic computation with error comparison.
The Lagrange remainder Rn(x) = f(n+1)(c)/(n+1)! ยท (xโˆ’a)n+1 bounds the error between f(x) and its nth-degree Taylor polynomial, where c is some value between a and x. To get an upper bound, find the maximum of |f(n+1)| on the interval. Use Error Bound mode to compute it automatically.
Yes. Click Print Worksheet for 1,000+ practice problems across 6 question types: expansion, binomial series, nth derivative, limits, integral approximation, and error bounds. Filter by type and difficulty (basic, medium, hard, scholar). Each worksheet is randomly generated with a full answer key โ€” ideal for exam prep and classroom use.
6 categories: series expansion (Taylor / Maclaurin polynomial), binomial series, nth derivative via series, limit evaluation by series substitution, definite integral approximation, and Lagrange error bound. Each problem has 4 difficulty levels from basic to scholar-level, with full LaTeX-rendered answers.
Yes โ€” 100% free, no signup, no limits. Step-by-step derivative calculations, interactive convergence graph with term slider, radius of convergence analysis, printable practice worksheets with answer keys, Python compiler, LaTeX copy, and shareable URLs. All computation runs in your browser.