Which of the following statements are true?

Question

Which of the following statements are true?

1. If the power series  converges for x = -2, then it also converges for x = 2.
2. If the power series  converges for x = -3, then it also converges for x = -2.
3. If the interval of convergence for  is (-1, 1), then the interval of convergence for  is (2, 3)

A	I only
B	I and II
C	II only
D	II and III

Answer 1

Answer: C

1. If ∑ a_nxⁿ is convergent at x = −2, then the radius of convergence R ≥ 2. If R = 2, then it is still possible that the series is divergent at the endpoint x = 2. A counterexample is ∑ a_nxⁿ = ∑ xⁿ ⁄ (n2ⁿ).
2. If ∑ a_nxⁿ is convergent at x = −3, then the radius of convergence R ≥ 3, so x = −2 lies inside the interval of convergence and thus the series is convergent at x = −2.
3. If ∑ a_nxⁿ has interval of convergence (−1, 1), then the interval of convergence of ∑ a_n(x − 2)ⁿ is given by −1 < x - 2 < 1 or 1 < x < 3.