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 ?
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 ?
Updated On: Oct 1, 2024
- \(\sum^{\infin}_{n=1}(-1)^{n+1}a_n\) is a convergent series
- \(\sum^{\infin}_{n=1}\frac{(-1)^{n+1}}{n}(a_1+a_2+...+a_n)\) is a convergent series
- The radius of convergence of the power series \(\sum^{\infin}_{n=1}a_nx^n \text{ is }\frac{1}{3}\)
- \(\sum^{\infin}_{n=1}a_n\sin\frac{1}{a_n}\) is a convergent series
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The Correct Option is A, B, D
Solution and Explanation
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.
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