10.11 Radius and Interval of Convergence of Power Series

Cards (47)

  • What is the definition of a power series centered at \( c \)?
    ∑n=0∞an(x−c)n\sum_{n = 0}^{\infty} a_{n}(x - c)^{n}
  • The variable in a power series determines its convergence.

    True
  • What are the coefficients in the power series \(\sum_{n=0}^{\infty} \frac{(x-2)^n}{n!}\)?
    an=a_{n} =1n! \frac{1}{n!}
  • What is the purpose of the Ratio Test for power series?
    Determine convergence
  • In the Ratio Test, the second step is to simplify the ratio.
  • The radius of convergence determines how far \( x \) can be from the center for convergence.
    True
  • In the third step of the Ratio Test, you must evaluate the limit as \( n \) approaches infinity.
  • What is the Ratio Test used for in the context of power series?
    Determining convergence
  • What is the radius of convergence of a power series?
    Maximum distance for convergence
  • Steps to find the radius of convergence using the Ratio Test
    1️⃣ Set up the ratio of consecutive terms
    2️⃣ Simplify the ratio
    3️⃣ Evaluate the limit as n approaches infinity
    4️⃣ Interpret the result to determine convergence or divergence
  • The full interval of convergence includes the lower and upper endpoints if the series converges at both.

    True
  • A power series centered at \( c \) is an infinite series of the form \(\sum_{n=0}^{\infty} a_n(x-c)^n\), where \( c \) is the center
  • Match the component of a power series with its description:
    Center ↔️ Fixed point around which the series is centered
    Variable ↔️ Determines the convergence of the series
    Coefficients ↔️ Constant values multiplying each term
  • The set of values for which a power series converges is known as the interval of convergence.
  • Steps for applying the ratio test to determine convergence:
    1️⃣ Set up the ratio of consecutive terms
    2️⃣ Simplify the ratio
    3️⃣ Evaluate the limit as n approaches infinity
    4️⃣ Interpret the results
  • What is the radius of convergence of a power series?
    Maximum distance for convergence
  • What test is used to find the radius of convergence of a power series?
    Ratio test
  • If the limit in the ratio test is less than 1, the series converges, and the radius of convergence is |x-c|.

    True
  • What happens if the limit in the ratio test equals 1?
    The test is inconclusive
  • The radius of convergence for the power series \(\sum_{n=0}^{\infty} \frac{x^n}{n!}\) is infinite.

    True
  • Steps for testing the endpoints of the interval of convergence
    1️⃣ Plug each endpoint value into the power series
    2️⃣ Use convergence tests to check convergence at each endpoint
    3️⃣ Include or exclude endpoints based on convergence
  • The harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\) diverges at x = 1.

    True
  • Steps for finding the interval of convergence
    1️⃣ Use the ratio test to find the radius of convergence
    2️⃣ Test the endpoints by plugging their values into the series
    3️⃣ Use convergence tests to determine endpoint convergence
    4️⃣ Write the interval of convergence, including convergent endpoints
  • The fixed point around which a power series is centered is called the center
  • Match the component of a power series with its symbol:
    Center ↔️ \( c \)
    Variable ↔️ \( x \)
    Coefficients ↔️ \( a_n \)
  • The set of values for \( x \) for which a power series converges is called the interval of convergence.
  • If the limit in the Ratio Test is greater than 1, the power series diverges.
    True
  • Steps to apply the Ratio Test
    1️⃣ Set up the ratio
    2️⃣ Simplify the ratio
    3️⃣ Evaluate the limit
    4️⃣ Interpret the results
  • What does the Ratio Test compare to determine convergence of a power series?
    Consecutive terms
  • For what values of \( x \) does the power series \(\sum_{n=0}^{\infty} \frac{x^n}{n!}\) converge?
    All values of \( x \)
  • In the Ratio Test, the ratio of consecutive terms is simplified by canceling common factors
  • If the limit in the Ratio Test is greater than 1, the radius of convergence is 0.

    True
  • What are the endpoints of the interval of convergence called?
    Critical points
  • In a power series, the center refers to a fixed point
  • Match the components of a power series with their descriptions:
    Center ↔️ The fixed point around which the series is centered
    Variable ↔️ The value that determines convergence
    Coefficients ↔️ The constant values multiplying each term
  • What is the center of the power series \(\sum_{n=0}^{\infty} \frac{(x-2)^n}{n!}\)?
    2
  • If the limit in the ratio test is greater than 1, the power series converges.
    False
  • The power series \(\sum_{n=0}^{\infty} \frac{x^n}{n!}\) converges for all values of \( x \) because the limit in the ratio test is 0.
  • If the limit in the ratio test is less than 1, the radius of convergence is |x-c|.

    True
  • Steps for using the ratio test to find the radius of convergence
    1️⃣ Set up the ratio of the (n+1)th term to the nth term
    2️⃣ Simplify the ratio by canceling common factors
    3️⃣ Evaluate the limit as n approaches infinity
    4️⃣ Interpret the result based on the limit's value