1. Let's start by stating the problem: We want to understand what Taylor and Maclaurin series are and how they are used to approximate functions. 2. A Taylor series of a function $f(x)$ centered at $a$ is an infinite sum of terms calculated from the derivatives of $f$ at $a$. The formula is: $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots$$ Here, $f^{(n)}(a)$ denotes the $n$th derivative of $f$ evaluated at $a$, and $n!$ is the factorial of $n$. 3. The Maclaurin series is a special case of the Taylor series centered at $a=0$. Its formula is: $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots$$ 4. Important rules: - The function must be infinitely differentiable at the point $a$ for the series to exist. - The series approximates the function near $a$; the more terms you include, the better the approximation. - Factorials grow quickly, which affects the size of terms. 5. Let's see an example: Find the Maclaurin series for $e^x$. - The derivatives of $e^x$ are all $e^x$, so at $x=0$, $f^{(n)}(0) = 1$ for all $n$. - Substitute into the formula: $$e^x = \sum_{n=0}^\infty \frac{1}{n!} x^n = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$ 6. This series converges to $e^x$ for all real $x$. 7. In summary, Taylor and Maclaurin series let us express complicated functions as infinite polynomials, making them easier to analyze and compute.