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The sequence {n√0.003} converges to A. 0 B. 1 C. 0.003 D. Infinity?
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The sequence {n√0.003} converges to A. 0 B. 1 C. 0.003 D. Infinity?
Sequence Convergence Analysis:
The given sequence is {n√0.003}, where n is a positive integer. We need to determine its convergence.
Definition of Convergence:
A sequence converges to a limit A if, for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between the nth term of the sequence and A is less than ε.
Analysis of the Given Sequence:
As n approaches infinity, the term n√0.003 approaches infinity as well. This is because the square root of 0.003 is a constant value less than 1, and multiplying it by an increasing positive integer n results in a value that grows without bound.
Conclusion:
Since the sequence {n√0.003} grows without bound as n approaches infinity, the sequence does not converge to a finite limit. Therefore, the correct answer is:
Option D. Infinity
The given sequence is {n√0.003}, where n is a positive integer. We need to determine its convergence.
Definition of Convergence:
A sequence converges to a limit A if, for every positive real number ε, there exists a positive integer N such that for all n greater than N, the absolute difference between the nth term of the sequence and A is less than ε.
Analysis of the Given Sequence:
As n approaches infinity, the term n√0.003 approaches infinity as well. This is because the square root of 0.003 is a constant value less than 1, and multiplying it by an increasing positive integer n results in a value that grows without bound.
Conclusion:
Since the sequence {n√0.003} grows without bound as n approaches infinity, the sequence does not converge to a finite limit. Therefore, the correct answer is:
Option D. Infinity
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The sequence {n√0.003} converges to A. 0 B. 1 C. 0.003 D. Infinity?
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The sequence {n√0.003} converges to A. 0 B. 1 C. 0.003 D. Infinity? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The sequence {n√0.003} converges to A. 0 B. 1 C. 0.003 D. Infinity? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sequence {n√0.003} converges to A. 0 B. 1 C. 0.003 D. Infinity?.
The sequence {n√0.003} converges to A. 0 B. 1 C. 0.003 D. Infinity? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The sequence {n√0.003} converges to A. 0 B. 1 C. 0.003 D. Infinity? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sequence {n√0.003} converges to A. 0 B. 1 C. 0.003 D. Infinity?.
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