Overview
A Taylor polynomial approximates a function near a point \( a \). Its error term is given by:
\( E_n(f) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1} \)
This term helps you estimate the error between the function and its polynomial approximation.
Example
For \( f(x) = xe^x \) centered at \( a = 1 \), the Taylor polynomial of order 2 is:
\( T_2(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2}(x-1)^2 \)
Evaluated as:
\( T_2(x) = e + 2e(x-1) + \frac{3}{2}e(x-1)^2 \)
The corresponding error term is:
\( E_2(x) = \frac{f'''(\xi)}{6}(x-1)^3 \)
with \( \xi \) between \( x \) and 1.
Maclaurin Identities
Maclaurin series are Taylor series expansions about 0 that yield useful identities for approximating functions. Some key examples are:
- Geometric Series: \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \) for \( |x| < 1 \).
- Exponential: \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \).
- Sine: \( \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} \).
- Cosine: \( \cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} \).
- Logarithm: \( \ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \) for \( |x| < 1 \).