📈 Calculus & Power Series Expansion
Taylor & Maclaurin Series Calculator
Compute $N^{\text{th}}$-order Taylor and Maclaurin polynomial expansions ($f(x) \approx \sum \frac{f^{(n)}(a)}{n!}(x-a)^n$) and compare approximation residuals against exact analytic values.
Expansion Parameters
Taylor Polynomial Approximation P_N(x)
2.71667
Generated Polynomial Formula $P_N(x)$:
$$P_5(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120}$$
Absolute Error ($|f(x) - P_N|$)
0.001615
Relative Accuracy
99.94% Accurate
💡 Lagrange Remainder Bound:
$$R_5(x) \le \frac{M}{(N+1)!} |x|^{N+1} = \frac{e^1}{720} \approx 0.00377$$
The Taylor Series Formula
Brook Taylor formulated the expansion of infinitely differentiable functions:
$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots$$
- Exponential: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$$
- Sine: $$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots$$
- Cosine: $$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots$$