10.14 Representing Functions by Maclaurin Series

• What is the general form of a power series?

  $\sum_{n = 0}^{\infty} c_{n}(x - a)^{n}$
• The convergence of a power series depends on the values of x
• A Taylor series is centered at $x =$$a$.
• What is the center of a Maclaurin series?

  $x =$$0$
• In a Taylor series, the coefficients are \frac{f^{(n)}(a)}{n!}
• The Maclaurin series is a special case of the Taylor series centered at $x =$$0$.
• What is the formula for the Maclaurin series of a function $f(x)$?


  $f(x) =$$\sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!}x^{n}$
• In the Maclaurin series formula, $f^{(n)}(0)$ represents the n-th derivative of $f$ evaluated at $x =$$0$.
• Steps to find the Maclaurin series of a function

  1️⃣ Compute the derivatives of $f(x)$.

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

  3️⃣ Form the Maclaurin series using the formula.
• The Maclaurin series is always centered at $x =$$0$.
• What is the Maclaurin series for $e^{x}$?


  $\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}$
• The radius of convergence for the Maclaurin series of $e^{x}$ is \infty
• Match the function with its Maclaurin series:

  $\sin x$ ↔️ $\sum_{n = 0}^{\infty} \frac{( - 1)^{n} x^{2n + 1}}{(2n + 1)!}$

  $\cos x$ ↔️ $\sum_{n = 0}^{\infty} \frac{( - 1)^{n} x^{2n}}{(2n)!}$

  $\frac{1}{1 - x}$ ↔️ $\sum_{n = 0}^{\infty} x^{n}$
• Steps to derive the Maclaurin series for $e^{x}$


  1️⃣ Find the $n$-th derivative of $e^{x}$.

  2️⃣ Evaluate the derivative at $x =$$0$.

  3️⃣ Substitute the values into the Maclaurin series formula.
• All derivatives of $e^{x}$ are equal to $e^{x}$.
• What is the general form of a power series?

  $\sum_{n = 0}^{\infty} c_{n}(x - a)^{n}$
• The convergence of a power series depends on the value of x
• What is the center of a Taylor series?

  $a$
• Maclaurin series represent functions using derivatives evaluated at the origin.
• What is the general form of a power series?

  $\sum_{n = 0}^{\infty} c_{n}(x - a)^{n}$
• The convergence of a power series depends on the value of x
• The Taylor series is a special case of the power series.
• What is the formula for the coefficients in a Taylor series?

  \frac{f^{(n)}(a)}{n!}</latex>
• The center of a Maclaurin series is always 0
• What is the defining characteristic of a Maclaurin series?

  Centered at $x =$$0$
• $f^{(n)}(0)$ in the Maclaurin series formula denotes then</latex>-th derivative of $f$ evaluated at $x =$$0$
• Steps to find the Maclaurin series of a function

  1️⃣ Compute Derivatives

  2️⃣ Evaluate Derivatives at $x =$$0$

  3️⃣ Form the Maclaurin Series
• Match the type of series with its center:

  Taylor Series ↔️ a

  Maclaurin Series ↔️ 0
• What is the general formula for the Maclaurin series of a function $f(x)$?


  $\sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}$
• The radius of convergence for the Maclaurin series of $e^{x}$ is $\infty$
• What is the Maclaurin series for $e^{x}$?


  $\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}$
• Techniques for constructing Maclaurin series using algebraic operations

  1️⃣ Substitution

  2️⃣ Addition/Subtraction

  3️⃣ Multiplication by x

  4️⃣ Differentiation/Integration
• What is the result of substituting $- x^{2}$ into the Maclaurin series for $e^{x}$?


  $\sum_{n = 0}^{\infty} \frac{( - x^{2})^{n}}{n!}$
• The Maclaurin series for $x e^{x}$ is obtained by multiplying the series for $e^{x}$ by x
• Maclaurin series for new functions can be obtained by differentiating or integrating known series term-by-term.
• What is the Maclaurin series for $\cos x$ obtained by differentiating the series for $\sin x$?


  $\sum_{n = 0}^{\infty} \frac{( - 1)^{n}}{(2n)!} x^{2n}$
• Maclaurin series for new functions can be obtained by differentiating or integrating known series term-by-term
• The constant of integration in the integral of a Maclaurin series is denoted by C
• Steps to find the derivative of the Maclaurin series for $\sin x$


  1️⃣ Write the Maclaurin series for $\sin x$

  2️⃣ Differentiate term-by-term

  3️⃣ Simplify the resulting series
• Match the function with its derived Maclaurin series:

  $\sin x$ ↔️ $\cos x =$$\sum_{n = 0}^{\infty} \frac{( - 1)^{n}}{(2n)!} x^{2n}$

  $\frac{1}{1 - x}$ ↔️ $- \ln(1 - x) =$$\sum_{n = 0}^{\infty} \frac{x^{n + 1}}{n + 1} +$$C$