Question

Let R1 and R2 be the radii of convergence of the power series $\sum\limits_{n=1}^{\infin}(-1)^nx^{n-1}$ and $\sum\limits_{n=1}^{\infin}(-1)^n\frac{x^{n+1}}{n(n+1)}$, respectively. Then

• A. R1 = R2
• B. R2 > 1
• C. $\sum\limits_{n=1}^{\infin}(-1)^nx^{n-1}$ converges for all x ∈ [−1, 1]
• D. $\sum\limits_{n=1}^{\infin}(-1)^n\frac{x^{n+1}}{n(n+1)}$ converges for all x ∈ [−1, 1]

Answer 1

Answer: (A), (D)

R1 = R2

Answer 2

The correct option is (A) : R1 = R2 and (D) : $\sum\limits_{n=1}^{\infin}(-1)^n\frac{x^{n+1}}{n(n+1)}$ converges for all x ∈ [−1, 1].