10.13 Finding Maclaurin Series for a Function

• The Maclaurin Series is a special case of the Taylor Series centered at 0
• The formula for the Maclaurin Series is $f(x) =$$\sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}$
• Match the components of the Maclaurin Series formula with their descriptions:

  $f^{(n)}(0)$ ↔️ The nth derivative of f(x) evaluated at x = 0

  $n!$ ↔️ The factorial of n

  $x^{n}$ ↔️ The power of x
• The Maclaurin Series for $f(x) =$$e^{x}$ is $e^{x}$
• The Maclaurin Series is centered at $a =$$0$ and represents a function as an infinite sum of powers of $x$
• Steps to find the Maclaurin Series:

  1️⃣ Calculate the first few derivatives

  2️⃣ Evaluate derivatives at $x =$$0$

  3️⃣ Summarize derivatives and values in a table
• The Maclaurin Series for f(x) = \sin(x)</latex> includes derivatives such as $\cos(x)$ and $- \sin(x)$
• The Maclaurin Series is a special case of the Taylor Series centered at $a =$$0$.
• The Maclaurin Series for $f(x) =$$e^{x}$ is 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots</latex>
• In the Maclaurin Series formula, $f^{(n)}(0)$ represents the derivative of $f(x)$ evaluated at $x =$$0$
• The Maclaurin Series is an infinite sum of powers of $x$ centered at $a =$$0$.
• What is the Maclaurin Series for e^{x}?

  $\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}$
• Steps to find the Maclaurin Series for a function

  1️⃣ Calculate the first few derivatives

  2️⃣ Evaluate the derivatives at $x =$$0$

  3️⃣ List the derivatives and their values
• What is the first derivative of $\cos(x)$?


  $- \sin(x)$
• The value of $- \sin(x)$ at x =0</latex> is 0
• The second derivative of $\cos(x)$ is $- \cos(x)$.
• Match the derivative of \cos(x)</latex> with its value at $x =$$0$:


  $\cos(x)$ ↔️ 1

  $- \sin(x)$ ↔️ 0

  $- \cos(x)$ ↔️ -1
• What is the general formula for the Maclaurin Series of a function $f(x)$?


  $f(x) =$$\sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}$
• The Maclaurin Series for $f(x) =$$e^{x}$ has all derivatives equal to 1
• To find the Maclaurin Series for a function, you must evaluate its derivatives at $x =$$0$.
• What is the Maclaurin Series for $\cos(x)$?


  $1 - \frac{x^{2}}{2!} +$$\frac{x^{4}}{4!} - \cdots$
• The Maclaurin Series for $\cos(x)$ is an example of an infinite sum
• Match the term with its definition:

  Maclaurin Series ↔️ Special case of Taylor Series at $a =$$0$

  Derivative ↔️ Rate of change of a function

  Factorial ↔️ Product of integers from 1 to n
• The n!</latex> in the Maclaurin Series formula represents the factorial of $n$.
• What is the general formula for the Maclaurin Series?

  $f(x) =$$\sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}$
• The derivatives of e^{x}</latex> are always equal to e^{x}
• The value of $f^{(n)}(0)$ for $f(x) =$$e^{x}$ is always 1.
• To find the Maclaurin Series for a function, we need to calculate its derivatives and evaluate them at x = 0
• Steps to find the Maclaurin Series for a function

  1️⃣ Calculate the first few derivatives

  2️⃣ Evaluate the derivatives at $x =$$0$

  3️⃣ Summarize derivatives and values in a table
• What is the function used as an example for calculating derivatives in the study material?

  $f(x) =$$e^{x}$
• The values $f^{(n)}(0)$ are crucial for determining the coefficients in the Maclaurin Series formula
• What is the Maclaurin Series formula?

  f(x) = \sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}</latex>
• Steps to evaluate derivatives at $x =$$0$


  1️⃣ Calculate the derivatives of the function

  2️⃣ Substitute $x =$$0$ into each derivative

  3️⃣ Summarize the values in a table
• An example of a function used to evaluate derivatives at x = 0</latex> is f(x) = \sin(x)
• Match the step with its description in the process of plugging derivative values into the Maclaurin Series formula

  List derivatives and values ↔️ Summarize the derivatives and their values at $x =$$0$

  Substitute into formula ↔️ Replace $f^{(n)}(0)$ in the formula

  Simplify the expression ↔️ Combine like terms and reduce coefficients
• What is the Maclaurin Series for $f(x) =$$e^{x}$?


  \sum_{n = 0}^{\infty} \frac{1}{n!} x^{n}</latex>
• The interval of convergence of a power series is the set of all $x$ values for which the series converges
• Match the type of interval of convergence with its range and behavior

  Infinite ↔️ $- \infty < x < \infty$ and converges for all real numbers

  Finite ↔️ $a - R < x < a +$$R$ and converges within radius $R$

  Single Point ↔️ $x =$$a$ and converges only at the center $a$