Question

Let {an}n≥1 be a sequence of real numbers such that $a_n=\frac{1}{3^n}$ for all n ≥ 1. Then which of the following statements is/are true ?

• A. $\sum^{\infin}_{n=1}(-1)^{n+1}a_n$ is a convergent series
• B. $\sum^{\infin}_{n=1}\frac{(-1)^{n+1}}{n}(a_1+a_2+...+a_n)$ is a convergent series
• C. The radius of convergence of the power series $\sum^{\infin}_{n=1}a_nx^n \text{ is }\frac{1}{3}$
• D. $\sum^{\infin}_{n=1}a_n\sin\frac{1}{a_n}$ is a convergent series

Answer 1

Answer: (A), (B), (D)

$\sum^{\infin}_{n=1}(-1)^{n+1}a_n$ is a convergent series

Answer 2

The correct option is (A) : $\sum^{\infin}_{n=1}(-1)^{n+1}a_n$ is a convergent series, (B) : $\sum^{\infin}_{n=1}\frac{(-1)^{n+1}}{n}(a_1+a_2+...+a_n)$ is a convergent series and (D) : $\sum^{\infin}_{n=1}a_n\sin\frac{1}{a_n}$ is a convergent series.