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How to find the limit sequence?

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Last updated date: 29th Feb 2024
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Answer
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Hint: The limit of the sequence is the value the sequence that is obtained when the number of terms value goes to infinity, the limit that approaches to a fixed value is called convergent sequence and the sequence that does not come to fixed value is called divergent. All the sequences do not have a limit also if the sequence has a limit then it must be unique.

Complete Step By Step solution:
If the sequence is such that, the higher terms are smaller in magnitude and the difference of the consecutive terms is decreased as the order of the terms is increased then the limit point is obtained. The example for it is \[{\left\{ {\dfrac{1}{n}} \right\}_n} = \left\{ {1,\dfrac{1}{2},\dfrac{1}{3},...} \right\}\] in this case the limit is zero as the higher the terms are the closer the value get is 0 and for large value of n the expression is infinity. Then \[\mathop {\lim }\limits_{n \to \infty } \dfrac{1}{n} = 0\] this is known as the limit point. The two things to be answered, the limit point is unique for every sequence, the second limit point may not be the member of the sequence as it is in this that is zero.
There is also other type of sequence that is called the oscillating sequence represented as,
\[{\left\{ {\left\{ {{{\left( { - 1} \right)}^n}} \right\}} \right\}_n} = \left\{ { - 1,1, - 1,1,...} \right\}\]
The above sequence does not converge or diverge and is called oscillatory.

Note:
As we know that there are some other type of sequence in which the consecutive term is increased in the magnitude for the higher value of n and the difference between the two terms is increased and the sequence that diverges altogether, then, \[{\left\{ n \right\}_n} = \left\{ {1,2,3} \right\}\], here the limit n tends to infinity.