10.14 Representing Functions by Maclaurin Series

Cards (72)

  • The Maclaurin series converges to \( f(x) \) if the interval of convergence includes \( x \)

    True
  • What is the \( n \)-th derivative of \( f(x) \) evaluated at \( x = 0 \) called in the Maclaurin series formula?
    \( f^{(n)}(0) \)
  • The Maclaurin series for \( \sin x \) is \( \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \) with an interval of convergence of (-\infty, \infty)
  • The Maclaurin series is a special type of Taylor series centered at \( a = 0
  • Match the components of a Maclaurin series with their descriptions:
    \( f^{(n)}(0) \) ↔️ The \( n \)-th derivative of \( f(x) \) evaluated at \( x = 0 \)
    \( n! \) ↔️ The factorial of \( n \)
    \( x^n \) ↔️ The \( n \)-th power of \( x \)
  • What is the Maclaurin series for \( e^x \)?
    \sum_{n=0}^{\infty} \frac{x^n}{n!}</latex>
  • The interval of convergence for the Maclaurin series of \( \cos x \) is \( (-\infty, \infty) \).

    True
  • Find the Maclaurin series for \( f(x) = \cos x \).
    cos⁡x=\cos x =∑n=0∞(−1)nx2n(2n)! \sum_{n = 0}^{\infty} \frac{( - 1)^{n} x^{2n}}{(2n)!}
  • What is a Maclaurin series a special type of?
    Taylor series
  • What determines the convergence of a Maclaurin series?
    Interval of convergence
  • Match the function with its Maclaurin series:
    exe^{x} ↔️ ∑n=0∞xnn!\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}
    sin⁡x\sin x ↔️ ∑n=0∞(−1)nx2n+1(2n+1)!\sum_{n = 0}^{\infty} \frac{( - 1)^{n} x^{2n + 1}}{(2n + 1)!}
    cos⁡x\cos x ↔️ ∑n=0∞(−1)nx2n(2n)!\sum_{n = 0}^{\infty} \frac{( - 1)^{n} x^{2n}}{(2n)!}
  • The Maclaurin series converges within its interval
  • Match the function with its interval of convergence:
    exe^{x} ↔️ (−∞,∞)( - \infty, \infty)
    11−x\frac{1}{1 - x} ↔️ (−1,1)( - 1, 1)
  • What is a Maclaurin series centered at?
    a = 0
  • Match the function with its Maclaurin series:
    e^x ↔️ \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \)
    \sin x ↔️ \( \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \)
    \cos x ↔️ \( \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \)
  • The interval of convergence determines the range of \( x \) values for which the Maclaurin series converges to \( f(x) \)

    True
  • What is a Maclaurin series a special case of?
    Taylor series
  • Arrange the components of a Maclaurin series in their order of appearance in the formula:
    1️⃣ \( f^{(n)}(0) \)
    2️⃣ \( n! \)
    3️⃣ \( x^n \)
  • What is the general formula for a Maclaurin series?
    f(x)=f(x) =∑n=0∞f(n)(0)n!xn \sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}
  • The Maclaurin series converges to \( f(x) \) if \( x \) falls within the interval of convergence.

    True
  • The Maclaurin series for \( \sin x \) is \( \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \)
  • Steps to find the Maclaurin series for a function \( f(x) \):
    1️⃣ Calculate Derivatives
    2️⃣ Identify a Pattern
    3️⃣ Generate the nth-Order Term
  • The general nth-order term for the Maclaurin series of \( \cos x \) is \( \frac{(-1)^n x^{2n}}{(2n)!} \)
  • A Maclaurin series represents a function as an infinite sum of terms
  • The Maclaurin series for exe^{x} converges for all real numbers.

    True
  • What is the term f(n)(0)f^{(n)}(0) in the Maclaurin series formula?

    n-th derivative at 0
  • The Maclaurin series for sin⁡x\sin x has an interval of convergence of (−∞,∞)( - \infty, \infty).

    True
  • What is the interval of convergence for the Maclaurin series of cos⁡x\cos x?

    (−∞,∞)( - \infty, \infty)
  • The Maclaurin series for \( e^x \) is \(\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}\) with an interval of convergence of \(( - \infty, \infty)\).

    True
  • The Maclaurin series for \( \frac{1}{1-x} \) is \(\sum_{n=0}^{\infty} x^n\) with an interval of convergence of \(( - 1, 1)\).
  • To find the Maclaurin series for a function \( f(x) \), we determine its derivatives at \(x = 0\).
  • The Maclaurin series for \( \cos x \) is \(\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\).
  • The radius of convergence is given by \(R = \frac{1}{L}\).
  • The ratio test is used to determine the radius and interval of convergence
  • The radius of convergence is calculated as 1/L
  • The interval of convergence for the Maclaurin series of \( e^x \) is (-∞, ∞).
    True
  • What is the Maclaurin series for \( e^{-x^2} \)?
    ∑n=0∞(−1)nx2nn!\sum_{n = 0}^{\infty} \frac{( - 1)^{n} x^{2n}}{n!}
  • A Maclaurin series is a Taylor series centered at \( a = 0 \).

    True
  • The convergence of a Maclaurin series is determined by its interval of convergence.

    True
  • What is the purpose of Maclaurin series in mathematics?
    Simplify complex calculations