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A real number P is cluster point o f a sequence < a
n
> if for any e > 0,- a)an ∈ ]P - ε, P + ε [ for infinitely many values of n
- b)an ∈ [P - ε, P + ε ] for infinitely many values of n
- c)an ∈ ]P - ε, P + ε [ for unique value of n
- d)an ∈ [P - ε, P + ε ] for unique value of n
Correct answer is option 'A'. Can you explain this answer?
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A real number P is cluster point o f a sequence < an> if for any...
A real number P is a cluster point of a sequence if there exists a subsequence of the sequence that converges to P. In other words, if we can find infinitely many terms of the sequence that are arbitrarily close to P.
To formally define a cluster point, let's denote the sequence as {a_n}, where n is a positive integer. Then, P is a cluster point of {a_n} if for every positive integer N, there exists a positive integer n>N such that |a_n - P| < ε="" for="" any="" positive="" real="" number="" />
In simpler terms, P is a cluster point if we can find terms of the sequence that get arbitrarily close to P, no matter how small of a distance we choose.
For example, consider the sequence {1/n}. The real number 0 is a cluster point of this sequence because we can always find terms of the sequence that are arbitrarily close to 0. If we choose ε=0.1, we can find terms like 1/10, 1/100, 1/1000, and so on, which are all within 0.1 distance from 0. Similarly, we can choose smaller ε values to find terms that are even closer to 0.
On the other hand, a number that is not a cluster point would be a number where we cannot find terms of the sequence that get arbitrarily close to it. For example, in the sequence {(-1)^n}, there are two cluster points, -1 and 1, because we can find terms that are arbitrarily close to these numbers. However, the number 0 is not a cluster point because no matter how small of a distance we choose, we cannot find terms of the sequence that get arbitrarily close to 0.
In summary, a real number P is a cluster point of a sequence if we can find a subsequence that converges to P, meaning that there are infinitely many terms of the sequence that are arbitrarily close to P.
To formally define a cluster point, let's denote the sequence as {a_n}, where n is a positive integer. Then, P is a cluster point of {a_n} if for every positive integer N, there exists a positive integer n>N such that |a_n - P| < ε="" for="" any="" positive="" real="" number="" />
In simpler terms, P is a cluster point if we can find terms of the sequence that get arbitrarily close to P, no matter how small of a distance we choose.
For example, consider the sequence {1/n}. The real number 0 is a cluster point of this sequence because we can always find terms of the sequence that are arbitrarily close to 0. If we choose ε=0.1, we can find terms like 1/10, 1/100, 1/1000, and so on, which are all within 0.1 distance from 0. Similarly, we can choose smaller ε values to find terms that are even closer to 0.
On the other hand, a number that is not a cluster point would be a number where we cannot find terms of the sequence that get arbitrarily close to it. For example, in the sequence {(-1)^n}, there are two cluster points, -1 and 1, because we can find terms that are arbitrarily close to these numbers. However, the number 0 is not a cluster point because no matter how small of a distance we choose, we cannot find terms of the sequence that get arbitrarily close to 0.
In summary, a real number P is a cluster point of a sequence if we can find a subsequence that converges to P, meaning that there are infinitely many terms of the sequence that are arbitrarily close to P.
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A real number P is cluster point o f a sequence < an> if for any e > 0,a)an ∈]P - ε, P + ε [ for infinitely many values of nb)an ∈ [P - ε, P + ε ]for infinitely many values of nc)an ∈]P - ε, P + ε [ for unique value of nd)an ∈ [P - ε, P + ε ]for unique value of nCorrect answer is option 'A'. Can you explain this answer?
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A real number P is cluster point o f a sequence < an> if for any e > 0,a)an ∈]P - ε, P + ε [ for infinitely many values of nb)an ∈ [P - ε, P + ε ]for infinitely many values of nc)an ∈]P - ε, P + ε [ for unique value of nd)an ∈ [P - ε, P + ε ]for unique value of nCorrect answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about A real number P is cluster point o f a sequence < an> if for any e > 0,a)an ∈]P - ε, P + ε [ for infinitely many values of nb)an ∈ [P - ε, P + ε ]for infinitely many values of nc)an ∈]P - ε, P + ε [ for unique value of nd)an ∈ [P - ε, P + ε ]for unique value of nCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A real number P is cluster point o f a sequence < an> if for any e > 0,a)an ∈]P - ε, P + ε [ for infinitely many values of nb)an ∈ [P - ε, P + ε ]for infinitely many values of nc)an ∈]P - ε, P + ε [ for unique value of nd)an ∈ [P - ε, P + ε ]for unique value of nCorrect answer is option 'A'. Can you explain this answer?.
A real number P is cluster point o f a sequence < an> if for any e > 0,a)an ∈]P - ε, P + ε [ for infinitely many values of nb)an ∈ [P - ε, P + ε ]for infinitely many values of nc)an ∈]P - ε, P + ε [ for unique value of nd)an ∈ [P - ε, P + ε ]for unique value of nCorrect answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about A real number P is cluster point o f a sequence < an> if for any e > 0,a)an ∈]P - ε, P + ε [ for infinitely many values of nb)an ∈ [P - ε, P + ε ]for infinitely many values of nc)an ∈]P - ε, P + ε [ for unique value of nd)an ∈ [P - ε, P + ε ]for unique value of nCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A real number P is cluster point o f a sequence < an> if for any e > 0,a)an ∈]P - ε, P + ε [ for infinitely many values of nb)an ∈ [P - ε, P + ε ]for infinitely many values of nc)an ∈]P - ε, P + ε [ for unique value of nd)an ∈ [P - ε, P + ε ]for unique value of nCorrect answer is option 'A'. Can you explain this answer?.
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