Taylor Series

Mathematical Principles & Theorems

Approximates smooth functions \(f(x) \in C^\infty\) near center point \(a\) via Taylor polynomials: \(P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k\). When \(a=0\), it is a Maclaurin series. Lagrange Remainder theorem bounds the approximation error: \(R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}\) for some \(\xi \in (a, x)\). Standard expansions: \(e^x = \sum_{k=0}^\infty \frac{x^k}{k!}\), \(\sin(x) = \sum_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)!}\), \(\cos(x) = \sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k)!}\), \(\frac{1}{1-x} = \sum_{k=0}^\infty x^k\).

Operating Instructions

• Select target function: \(\sin(x), \cos(x), e^x, \ln(1+x), \frac{1}{1-x}\) or custom function.
• Set the polynomial degree \(n\) using the precise numeric value box.
• Observe the polynomial curve \(P_n(x)\) dynamically converge toward \(f(x)\) as degree \(n\) increases.
• Inspect the error curve plot \(|f(x) - P_n(x)|\) and verified interval of convergence bounds.