Sine (Maclaurin)
\(\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\)
Calculatrice de série de Taylor
Générer des développements en séries de Taylor et de Maclaurin avec des approximations polynomiales étape par étape
Résultats
Entrez les valeurs et cliquez sur Calculer pour voir le résultat.
Theory & Formula
A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. When centered at x = 0, it's called a Maclaurin series.
Key properties:
- Polynomial Approximation: Taylor series approximate smooth functions using polynomials
- Convergence: Series converges within radius of convergence
- Error Term: Remainder R_n = f^(n+1)(c)(x-a)^(n+1)/(n+1)! for some c between a and x
- Applications: Used in numerical analysis, physics, engineering for approximations
- Common Series: sin(x), cos(x), e^x, ln(1+x), (1+x)^n all have simple Taylor expansions
\(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\)
Worked Examples
Cosine (Maclaurin)
\(\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\)
Exponential (Maclaurin)
\(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\)
Natural Log
\(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\)