Question

P: Suppose that \[ \sum_{n=0}^{\infty} a_n x^n \text{ converges at } x = -3 \text{ and diverges at } x = 6. \text{ Then } \sum_{n=0}^{\infty} (-1)^n a_n \text{ converges.} \] Q: The interval of convergence of the series \[ \sum_{n=2}^{\infty} \frac{(-1)^n x^n}{4^n \log_e n} \text{ is } [-4, 4]. \] Which of the following statements is TRUE?

• A. P is true and Q is true
• B. P is false and Q is false
• C. P is true and Q is false
• D. P is false and Q is true

Hint:

When analyzing power series, check the radius of convergence using the ratio or root test. The interval of convergence may not always include the endpoints, so verify convergence at those points separately.

Answer 1

The Correct Option is C

Step 1: Analyzing P.
The statement P tells us that the series \( \sum_{n=0}^{\infty} a_n x^n \) converges at \( x = -3 \) and diverges at \( x = 6 \). This means that the radius of convergence \( R \) of the power series is between 3 and 6. Since the series converges at \( x = -3 \) and diverges at \( x = 6 \), the interval of convergence for this series must be \( (-3, 6) \). Therefore, statement P is true.
Step 2: Analyzing Q.
The series in Q is given by \[ \sum_{n=2}^{\infty} \frac{(-1)^n x^n}{4^n \log_e n}. \] This is a power series with a general term \( \frac{(-1)^n x^n}{4^n \log_e n} \). The coefficient of \( x^n \) behaves like \( \frac{1}{4^n \log_e n} \), which decays faster than \( \frac{1}{4^n} \). Therefore, the radius of convergence is determined by \( \frac{1}{4} \), giving an interval of convergence of \( (-4, 4) \). However, the series is conditionally convergent at the endpoints \( x = \pm 4 \), and the interval of convergence is not exactly \( [-4, 4] \) because the behavior at the endpoints needs further investigation. Thus, statement Q is false.
Step 3: Final Answer.
The correct answer is (C) because statement P is true, and statement Q is false.