P:if $\sum_{n\ge0}a_nx^n$ converges at −3 and diverges at 6, then Σ(−1)ⁿa_n converges. Q:the convergence interval of $\sum_{n=2}^\infty(-1)^nx^n/(4^n\ln n)$ is [−4,4]. Choose truth values.
Convergence at −3 gives radius at least 3, so the series converges absolutely at x=−1, proving P. The second series has radius 4. At x=4 it is alternating 1/ln n and converges; at x=−4 it becomes Σ1/ln n and diverges (eventually ≥1/n). Its interval is (−4,4], so Q is false. **C**.
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