1. Let's start by stating the problem: We want to understand Taylor and Maclaurin series and how to analyze the pattern of derivatives $f^{(n)}(x)$ to find a general formula for the series. 2. The Taylor series of a function $f(x)$ centered at $a$ is given by the formula: $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x - a)^n$$ where $f^{(n)}(a)$ is the $n$th derivative of $f$ evaluated at $x=a$, and $n!$ is the factorial of $n$. 3. The Maclaurin series is a special case of the Taylor series centered at $a=0$: $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n$$ 4. To find the general term of the series, we need to analyze the pattern of the derivatives $f^{(n)}(x)$: - Compute the first few derivatives $f'(x), f''(x), f^{(3)}(x), \ldots$ - Evaluate each derivative at the center point $a$ (or 0 for Maclaurin). - Look for a pattern or formula that expresses $f^{(n)}(a)$ in terms of $n$. 5. Once the pattern is identified, substitute it back into the series formula to get the general term: $$\text{General term} = \frac{f^{(n)}(a)}{n!} (x - a)^n$$ 6. Example: For $f(x) = e^x$, all derivatives are $f^{(n)}(x) = e^x$, so $f^{(n)}(0) = 1$. The Maclaurin series is: $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$ 7. Important rules: - Factorials grow very fast, so terms often get smaller quickly. - The radius of convergence depends on the function. - Recognizing the pattern in derivatives is key to writing the series. In summary, analyzing $f^{(n)}(a)$ means computing derivatives, evaluating at $a$, and finding a formula for the $n$th derivative to write the series compactly.