Analysis question dealing with proving sequence and its limit
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Analysis question dealing with proving sequence and its limit

[From: ] [author: ] [Date: 13-10-23] [Hit: ]
-Set X(n) = n/(n + 1).Take ε > 0.Therefore lim X = 1.-Having a limit is what it means for a sequence to converge.......
PROVE that you have found the correct limit, or prove that the series is divergent.

Question: Is the sequence {n/n+1} convergent? If so, what is the limit.

Definition: A sequence {Xn} is said to converge to a number x element R, if for every e>0, there exists an M element N such that |Xn-x|=M. The number is said to be the limit of {Xn}. We write lim Xn=x as n approaches infinity.

What I have: I know that the sequence DIVERGES up to 1. Using the definition I get Let e>0, find M element N such that 1/M+1=M I get:

| n | | n - (n+1) | | 1 | 1
|------ - 1 | = |------------ | = |-------| <= ----- < e
| n+1 | | n+1 | | n+1 | M+1

Can I get someone knowledgeable in Mathematical Analysis help me with this problem please?

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Set X(n) = n/(n + 1).

Take ε > 0.
If N ∈ ℕ is the smallest integer with 1/N < ε then for n ≥ N we have
|X(n) - 1| = |n/(n + 1) - 1| = 1/(n + 1) < 1/N < ε.
Therefore lim X = 1.

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Having a limit is what it means for a sequence to converge.

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