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Series 2023

  1. The power series \(\sum_{n=1}^{\infty} \frac{(x-5)^n}{2^nn^2}\) has a radius of convergence of 2. At which of the following values of \(x\) can the alternating series test be used with this series to verify convergence at \(x\)?
    1. 6
    2. 4
    3. 2
    4. 0
    5. -1
  2. Which of the following statements about convergence of the series \(\sum_{n=1}^{\infty} \frac{1}{\ln(n+1)}\) is true?
    1. \(\sum_{n=1}^{\infty} \frac{1}{\ln(n+1)}\) converges by comparison to \(\sum_{n=1}^{\infty} \frac{1}{n}\).
    2. \(\sum_{n=1}^{\infty} \frac{1}{\ln(n+1)}\) converges by comparison to \(\sum_{n=1}^{\infty} \frac{1}{n^2}\).
    3. \(\sum_{n=1}^{\infty} \frac{1}{\ln(n+1)}\) diverges by comparison to \(\sum_{n=1}^{\infty} \frac{1}{n}\).
    4. \(\sum_{n=1}^{\infty} \frac{1}{\ln(n+1)}\) diverges by comparison to \(\sum_{n=1}^{\infty} \frac{1}{n^2}\).
  3. Which of the following series converges for all real numbers \(x\)?
    1. \(\sum_{n=1}^{\infty} \frac{x^n}{n}\)
    2. \(\sum_{n=1}^{\infty} \frac{x^n}{n^2}\)
    3. \(\sum_{n=1}^{\infty} \frac{x^n}{\sqrt{n}}\)
    4. \(\sum_{n=1}^{\infty} \frac{e^nx^n}{n!}\)
    5. \(\sum_{n=1}^{\infty} \frac{n!x^n}{e^n}\)
  4. Which of the following statements about the series \(\sum_{n=1}^{\infty} \frac{1}{2^n - n}\) is true?
    1. The series diverges by the \(n\)th term test
    2. The series diverges by limit comparison to the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\)
    3. The series converges by the \(n\)th term test
    4. The series converges by limit comparison to the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\)

  5. Which of the following series converge to 2?

    I. \(\sum_{n=1}^{\infty} \frac{2n}{n+3}\)
    II. \(\sum_{n=1}^{\infty} \frac{-8}{(-3)^n}\)
    III. \(\sum_{n=1}^{\infty} \frac{1}{2^n}\)

    1. I only
    2. II only
    3. III only
    4. I and III only
    5. II and III only

  6. Which of the following series converge?

    I. \(\sum_{n=1}^{\infty} \frac{8^n}{n!}\)
    II. \(\sum_{n=1}^{\infty} \frac{n!}{n^100}\)
    III. \(\sum_{n=1}^{\infty} \frac{n+1}{n(n+2)(n+3)}\)

    1. I only
    2. II only
    3. III only
    4. I and III only
    5. I, II, and III