The Taylor series expansion is a way to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. If \( f(x) \) is a function that is infinitely differentiable at a point \( a \), the Taylor series of \( f(x) \) about \( a \) is given by:
\[ f(x) = f(a) + f'(a) \frac{(x - a)}{1!} + f''(a) \frac{(x - a)^2}{2!} + f'''(a) \frac{(x - a)^3}{3!} + \cdots \]
In general, the \( n \)-th term of the series is:
\[ \frac{f^{(n)}(a)}{n!} (x - a)^n \]
where \( f^{(n)}(a) \) denotes the \( n \)-th derivative of \( f \) evaluated at \( x = a \), and \( n! \) denotes the factorial of \( n \).
When \( a = 0 \), the series is called the Maclaurin series:
\[ f(x) = f(0) + f'(0) \frac{x}{1!} + f''(0) \frac{x^2}{2!} + f'''(0) \frac{x^3}{3!} + \cdots \]
Taylor series can approximate functions near the point \( a \), and under certain conditions, they can represent the function exactly.
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