MATH1020U: Sequences and Series - Ratio and Root Tests
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Questions and Answers

What is the definition of a power series?

  • A series that involves factorials in each term
  • A series that converges for all x values
  • A series of the form $\sum_{n=0}^\infty c_nx^n$ (correct)
  • A series that converges to a single value
  • In a power series, what are the coefficients $c_n$ called?

  • Coefficients
  • Constants (correct)
  • Variables
  • X-factors
  • A power series in (x-a) or centered at/about 'a' has the form ___?

  • $\sum_{n=0}^\infty c_n(a-x)^n$
  • $\sum_{n=0}^\infty c_n(x-a)^n$ (correct)
  • $\sum_{n=0}^\infty c_n(x+a)^n$
  • $\sum_{n=0}^\infty c_nx^n$
  • At what value is the series $\sum_{n=0}^\infty n(x-6)^n$ centered?

    <p>$a = 6$</p> Signup and view all the answers

    What values of $x$ make the series $\sum_{n=0}^\infty n!x^n$ converge?

    <p>$x = 0$</p> Signup and view all the answers

    What is one of the strategies for determining if a given power series converges or diverges?

    <p>Applying the Root Test</p> Signup and view all the answers

    What is the radius of convergence for the series $\sum_{n=0}^\infty (x-2)^n$?

    <p>$2$</p> Signup and view all the answers

    For which values of $x$ does the series $\sum_{n=1}^\infty \frac{7}{n} \cos{n}$ converge?

    <p>$x &gt; 1$</p> Signup and view all the answers

    What is the radius of convergence for the series $\sum_{n=0}^\infty \frac{n^2}{7(n+9)}$?

    <p>$8$</p> Signup and view all the answers

    In the power series $\sum_{n=0}^\infty (8x-3)^n$, what value of R indicates the convergence boundary?

    <p>$5$</p> Signup and view all the answers

    For the series $\sum_{n=1}^\infty \frac{7}{n} \cos{n}$, why is it inconclusive at the endpoints of the interval of convergence?

    <p>Ratio and root tests are inconclusive there</p> Signup and view all the answers

    What does the Ratio Test conclude if $\lim_{n \to \infty} \frac{(-1)^n}{n+3}$ is less than 1?

    <p>The series is absolutely convergent</p> Signup and view all the answers

    When does the Ratio Test consider a series absolutely convergent?

    <p>$\lim_{n\to \infty} a_{n+1} = L &lt; 1$</p> Signup and view all the answers

    What does the Root Test conclude if $\lim_{n \to \infty} \sqrt[n]{7n}$ is less than 1?

    <p>The series is absolutely convergent</p> Signup and view all the answers

    When is the Root Test inconclusive?

    <p>$\lim_{n\to \infty} a_n = L = 1$</p> Signup and view all the answers

    In the Ratio Test, what conclusion can be drawn if $\lim_{n \to \infty} \frac{(-1)^n}{n+3}$ is greater than 1 or tends to infinity?

    <p>The series is divergent</p> Signup and view all the answers

    If $\lim_{n \to \infty} (7n)!$ tends to infinity in the Root Test, what conclusion can be made about the series?

    <p>The series is divergent</p> Signup and view all the answers

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