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L.sufiya Khanam posted an Question

October 08, 2020 • 13:02 pm

30 points

IIT JAM
Mathematics (MA)

Clarification

plz give some examples. for this. theorem. ??

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Piyush pachauri

you should try to construct example on your own that will help you. Take an = 2-(3/n) converges to 2 here m=1
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sir here Xn will be -1 ,1/2 ,1, 5/4........
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here Xn is -1 which is not greater than 0
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but in theorem Xn is greater than 0
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It says xn > 0 for all n>m
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after some natural no m , Xn should be greater than zero.
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ok sir
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Srinath Best Answer

This theorem states the simple fact that, when a sequence converges to a positive real number, all but a finite number of terms in the sequence has to be positive. That is, while approaching a positive real number at the limit, there are infinite terms of the sequence at the tail end, that are all positive. For example, consider the sequence, {n/(n+1)}. This sequence converges to 1, and there are infinite number of positive terms, (an) for any n. If this example isn't satisfactory, try (-1)^n/n
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sir for the second example limit is zero right?
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Yes, I forgot to type, 1 + ((-1)^n/n)
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but still it's not satisfactory as limit is 0, that is it's not greater than zero right?
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What you are saying is right. I wanted to type 1+ ((-1)^n/n), but I typed it without the 1, so it's limit becomes 0, which cannot be used in this scenario.
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okay sir, im telling that without 1 also it is not satisfactory right?
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Yes, it is not. Limit has to be greater than 0.
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ok sir 👍
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