Maclaurin Series Calculator
Compute the Maclaurin series (Taylor expansion about 0) of f(x) up to degree n.
Inputs
Enter function using x.
M(x) at x
-0.8412882325
f(x) exact
0.8414709848
Absolute error
1.6828 × 10⁰
Coefficients
c0=0, c1=-1, c2=0, c3=0.166667, c4=0, c5=-0.008333, c6=0
Step by step
Values used
f(x) = sin(x); Degree (n) = 7; Evaluate at x = 1
Maclaurin series
M(x) = Σ [f^(k)(0) / k!] · x^k, for k = 0 to n
M(x) at x
= -0.8412882325
f(x) exact
= 0.8414709848
Absolute error
= 1.6828
Coefficients
= c0=0, c1=-1, c2=0, c3=0.166667, c4=0, c5=-0.008333, c6=0
How it works
A Maclaurin series is a Taylor series centered at 0: M(x) = Σ f⁽ᵏ⁾(0)/k! · xᵏ. Common Maclaurin series include e^x = Σ xᵏ/k!, sin(x) = x - x³/3! + x⁵/5! - ..., cos(x) = 1 - x²/2! + x⁴/4! - .... Coefficients are computed numerically via finite differences.
Formula
Maclaurin series
M(x) = Σ [f^(k)(0) / k!] · x^k, for k = 0 to n
- f^(k)(0)
- k-th derivative at 0
- k!
- k factorial
- n
- Degree
Frequently Asked Questions
When should I use Maclaurin vs Taylor series?
Use Maclaurin when the function is well-behaved near 0. Use Taylor with a different center when you need accuracy far from the origin.
Why might the series converge slowly?
Functions with singularities or rapid oscillations near x=0 need many terms. The radius of convergence limits where the series is useful.