Maclaurin's Theorem
Generate Maclaurin Series for Functions
Determine the power series for simple trigonometric, logarithmic, and exponential functions.
Maclaurin's Theorem
Maclaurin's theorem is a special case of Taylor's theorem, where the power series expansion of a function $f(x)$ is about the point $x=0$.
The Maclaurin series for a function $f(x)$ is given by:
$$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \dots + \frac{f^{(n)}(0)}{n!}x^n + \dots$$
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$$
Conditions for Maclaurin's Series:
- The function $f(x)$ must be infinitely differentiable at $x=0$ (i.e., all its derivatives $f'(0), f''(0), f'''(0), \dots$ must exist).
- The Maclaurin series must converge to $f(x)$. This is often determined by checking if the remainder term $R_n(x)$ of Taylor's theorem (centered at 0) tends to zero as $n \to \infty$. For many common functions, this condition holds within a certain radius of convergence.
Evaluate Definite Integral using Maclaurin's Series
A definite integral $\int_a^b f(x)dx$ can be approximated by replacing $f(x)$ with its Maclaurin series and integrating term by term. This is particularly useful when $f(x)$ does not have an elementary antiderivative.
If $f(x) = \sum_{n=0}^{\infty} c_n x^n$, then $\int_a^b f(x)dx = \sum_{n=0}^{\infty} c_n \int_a^b x^n dx = \sum_{n=0}^{\infty} c_n \left[ \frac{x^{n+1}}{n+1} \right]_a^b$.
L'Hopital's Rule
L'Hopital's Rule is a method for finding the limit of a quotient of two functions that results in an indeterminate form such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
Statement: If $\lim_{x\to c} f(x) = \lim_{x\to c} g(x) = 0$ or $\pm\infty$, and if $\lim_{x\to c} \frac{f'(x)}{g'(x)}$ exists (and $g'(x) \neq 0$ near $c$, except possibly at $c$), then:
$$\lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)}$$
The rule can be applied repeatedly if the new limit is also indeterminate and the conditions are met.