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If a sequence of real number has a cluster points, then
- a)it is convergent
- b)it is divergent
- c)limit exist
- d)existence of limit not definite
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
If a sequence of real number has a cluster points, thena)it is converg...
Explanation:
To understand the answer to this question, let's first define what a cluster point is.
Cluster Point:
A cluster point of a sequence of real numbers is a real number such that for any neighborhood of the cluster point, there are infinitely many terms of the sequence in that neighborhood.
Now, let's analyze each option and see if it is true or false for a sequence with cluster points.
a) It is convergent:
This option is not necessarily true. A sequence with cluster points can be both convergent or divergent. Having cluster points does not guarantee that the sequence is convergent.
b) It is divergent:
This option is not necessarily true either. A sequence with cluster points can be both convergent or divergent. Having cluster points does not guarantee that the sequence is divergent.
c) Limit exists:
This option is also not necessarily true. A sequence with cluster points may or may not have a limit. The existence of cluster points does not imply the existence of a limit for the sequence.
d) Existence of limit not definite:
This option is correct. The existence of cluster points in a sequence means that the existence of a limit is not definite. It could exist or it could not exist. The presence of cluster points indicates that the behavior of the sequence is not determined and it is not possible to determine whether the sequence converges or diverges.
Therefore, the correct answer is option 'D', the existence of limit is not definite for a sequence with cluster points.
To understand the answer to this question, let's first define what a cluster point is.
Cluster Point:
A cluster point of a sequence of real numbers is a real number such that for any neighborhood of the cluster point, there are infinitely many terms of the sequence in that neighborhood.
Now, let's analyze each option and see if it is true or false for a sequence with cluster points.
a) It is convergent:
This option is not necessarily true. A sequence with cluster points can be both convergent or divergent. Having cluster points does not guarantee that the sequence is convergent.
b) It is divergent:
This option is not necessarily true either. A sequence with cluster points can be both convergent or divergent. Having cluster points does not guarantee that the sequence is divergent.
c) Limit exists:
This option is also not necessarily true. A sequence with cluster points may or may not have a limit. The existence of cluster points does not imply the existence of a limit for the sequence.
d) Existence of limit not definite:
This option is correct. The existence of cluster points in a sequence means that the existence of a limit is not definite. It could exist or it could not exist. The presence of cluster points indicates that the behavior of the sequence is not determined and it is not possible to determine whether the sequence converges or diverges.
Therefore, the correct answer is option 'D', the existence of limit is not definite for a sequence with cluster points.
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Community Answer
If a sequence of real number has a cluster points, thena)it is converg...
You can do it by discarding the options take sequence that has cluster point but not cgt Ex x_n= {2 when n is prime ; n when n is not prime }
for option b take x_n = 1/n has cluster point but nhi divergent similarly for c option take example as a option
for option b take x_n = 1/n has cluster point but nhi divergent similarly for c option take example as a option
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If a sequence of real number has a cluster points, thena)it is convergentb)it is divergentc)limit existd)existence of limit not definiteCorrect answer is option 'D'. Can you explain this answer?
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