Maclaurin Series Calculator
Compute the Maclaurin series (Taylor expansion about 0) of f(x) up to degree n.
About Maclaurin Series Calculator
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
Worked example
With the default values below, Maclaurin Series Calculator returns M(x) at x of -0.8412882325. Change any input above and every figure here is recalculated in your browser.
Inputs used
- f(x)
- sin(x)
- Degree (n)accepts 1–20
- 7
- Evaluate at x
- 1
Results
- 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
- 1Values used
- 2Maclaurin series
- 3M(x) at x= -0.8412882325
- 4f(x) exact= 0.8414709848
- 5Absolute error= 1.6828
- 6Coefficients= c0=0, c1=-1, c2=0, c3=0.166667, c4=0, c5=-0.008333, c6=0
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.