# [[Basic Maclaurin Expansions]] (Mathematics > Calculus) ## Definition A **Maclaurin series** is a Taylor series expansion of a function $f(x)$ about $x=0$. It expresses $f(x)$ as an infinite sum of powers of $x$, and is given by: $ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}x^k. $ Basic Maclaurin expansions refer to the standard series for common functions that are often memorized for quick reference. ## Key Concepts - **Taylor Series at Zero:** Maclaurin series are simply Taylor series with the center at $0$. - **Power Series Representation:** Functions are represented as an infinite sum of terms involving powers of $x$. - **Convergence:** Many standard series, such as those for $e^x$, $\sin(x)$, and $\cos(x)$, converge for all $x$, while others have restricted intervals. - **Substitution:** New series can be obtained by substituting expressions into these basic series. ## Important Properties 1. **Uniformity of Convergence:** For functions like $e^x$, $\sin(x)$, and $\cos(x)$, the series converge for all $x \in (-\infty, \infty)$. 2. **Alternating Series:** Functions like $\sin(x)$, $\cos(x)$, and $\ln(1+x)$ have alternating series with alternating signs. 3. **Factorial Growth:** The factorial in the denominator ensures that terms decrease rapidly, aiding convergence. ## Essential Formulas - **Exponential Function:** $ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots, \quad x \in (-\infty,\infty). $ - **Sine Function:** $ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots, \quad x \in (-\infty,\infty). $ - **Cosine Function:** $ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots, \quad x \in (-\infty,\infty). $ - **Natural Logarithm:** $ \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots, \quad x \in (-1,1]. $ - **Geometric Series:** $ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots, \quad x \in (-1,1). $ - **Alternate Geometric Series:** $ \frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots, \quad x \in (-1,1). $ ## Core Examples 1. **Basic Example:** To find the Maclaurin series for $e^{3x}$, substitute $3x$ for $x$ in the expansion for $e^x$: $ e^{3x} = 1 + (3x) + \frac{(3x)^2}{2!} + \frac{(3x)^3}{3!} + \cdots = 1 + 3x + \frac{9x^2}{2} + \frac{27x^3}{6} + \cdots. $ 2. **Advanced Application:** Use the Maclaurin series for $\ln(1+x)$ to approximate $\ln(1.1)$: $ \ln(1+x) \approx x - \frac{x^2}{2} + \frac{x^3}{3} \quad \text{with } x=0.1, $ so $ \ln(1.1) \approx 0.1 - \frac{0.01}{2} + \frac{0.001}{3} \approx 0.1 - 0.005 + 0.00033 \approx 0.09533. $ ## Related Theorems/Rules - **Taylor's Theorem:** Provides the remainder term which quantifies the error in approximating a function by a finite number of terms. - **Substitution Rule:** Allows the derivation of new series by replacing $x$ with another expression in the basic series. ## Common Pitfalls - **Forgetting the Interval of Convergence:** Each series has its own interval where it converges. - **Sign Errors:** Particularly with alternating series like $\sin(x)$, $\cos(x)$, and $\ln(1+x)$. - **Incorrect Substitution:** Not properly substituting the new variable (e.g., replacing $x$ with $3x$) can lead to mistakes. ## Related Topics - [[Taylor Series]] - [[Taylor Polynomial]] - [[Quadratic_Taylor_Polynomial]] - [[Infinite_Series]] - [[Convergence_of_Infinite_Series]] - [[Infinite Series Properties]] - [[Geometric Series]] - [[Infinite_Geometric_Series]] - [[Series Notation]] - [[Exponential Function and Its Integral]] - [[Derivative of the Exponential Function]] - [[Limit of Exponential Functions]] - [[Limits at Infinity for Exponential Functions]] - [[Derivatives of Trigonometric Functions]] - [[Limits of Sine and Cosine]] - [[Euler_s Formula]] - [[Binomial_Theorem]] - [[Linear_Approximation]] - [[LHopitalsRule]] - [[Limit of a Sequence]] - **Taylor Series:** General series expansion about any point. - **Power Series:** Representation of functions as an infinite sum of powers of $x$. - **Series Convergence Tests:** Methods to determine the interval and radius of convergence for power series. ## Quick Review Questions 1. How do you derive the Maclaurin series for a function like $e^{3x}$ using the basic expansion for $e^x$? 2. What is the interval of convergence for the Maclaurin series of $\frac{1}{1-x}$ and why? *** ![[Basic_Maclaurin_Expansions_visualization]]