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Taylor Series Expansion Calculator

Numerically compute Taylor coefficients and output the n-th order polynomial at center a.

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f(x) ≈ Σ f^(k)(a)/k! · (x−a)^k
e.g. exp(x) or sin(x)
expansion center
0 to 8

📖 Tutorial | Taylor Series

1. Theorem
Taylor formula: f(x)=Σ f^(k)(a)/k!·(x−a)^k. At a=0 it is a Maclaurin expansion. The first n+1 terms form the n-th Taylor polynomial.
2. Symbols
f^(k)(a)k-th derivative at a
c_k = f^(k)(a)/k!Taylor coefficient
acenter
norder
3. How it works
  • Numerically differentiate with central differences;
  • Divide by k! to get c_k;
  • Assemble c_k (x−a)^k;
  • Drop negligible terms.
4. Steps
  1. Enter f(x), e.g. exp(x);
  2. Set a=0, n=4;
  3. Click Calculate.
5. Example
Example: e^x to order 4 at 0.
Solution: c_k=1/k!, so P_4=1+x+x²/2+x³/6+x⁴/24. Coefficients [1,1,0.5,0.1667,0.0417] match.
6. Pitfalls
Numerical high-order derivatives are noisy; n>8 may be unstable;
The expansion is accurate near the center;
f must be smooth at a.

❓ FAQ | Taylor Series

What is a Maclaurin expansion?
The Taylor expansion about a=0, e.g. e^x=Σx^k/k!.
Why does it approximate?
It matches f and all its derivatives at a.
Is higher order always better?
Usually within the radius, but round-off grows; this tool limits n≤8.
What if a≠0?
Enter a directly; the tool writes (x−a)^k.
Common expansions?
e^x, sin x, cos x, ln(1+x), (1+x)^α.