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Maclaurin's Formula Calculator

Maclaurin series calculator. Enter f(x), order n and point to get the expansion coefficients, polynomial and remainder at x=0—for Taylor basics.

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f(x) = Σ f^(k)(0)/k!·x^k + R_n
Supports + - * / ^ and sin cos tan sqrt exp log pi e
Integer 1~6
Compare approximation here

📖 Tutorial｜Maclaurin's Formula

1. Definition
Maclaurin's formula is Taylor at a=0: f(x)=Σ f^(k)(0)/k!·x^k + R_n.
2. Symbols
SymbolMeaning
nOrder
f^(k)(0)k-th derivative at 0
R_nRemainder
3. How it works
  • We compute derivatives at the origin;
  • Build the Maclaurin polynomial;
  • Compare approximation and true value at x0.
4. Steps
  1. Enter f(x);
  2. Enter order n and point x0;
  3. Click Calculate.
5. Example
Example: sin x: x − x³/6. At x0=0.5, P≈0.47917 while sin0.5≈0.4794.
6. Pitfalls
Convergence is best near the origin;
High-order numerical error accumulates;
Memorize eˣ, sinx, cosx, ln(1+x) expansions.

❓ FAQ｜Maclaurin's Formula

Relation between Maclaurin and Taylor?
Maclaurin is Taylor with center a=0.
Common Maclaurin expansions?
eˣ=1+x+x²/2+…; sinx=x−x³/6+…; cosx=1−x²/2+…; ln(1+x)=x−x²/2+….
How to estimate the remainder?
Lagrange form R_n=f^(n+1)(ξ)/(n+1)!·x^(n+1) with ξ between 0 and x.
Why keep x0 small?
Maclaurin is accurate near the origin; error grows quickly far away.
Difference from the Taylor tool?
This one fixes a=0; the Taylor tool lets you set any center.