Prove that the sequence converges

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In summary, the given sequence x_n = (n^2 / (n^2+1) , 1/sqrt(n)) converges to the limit (1,0) as n approaches infinity. To prove this, we use the definition of a convergent sequence and choose an epsilon > 0. We then need to find some N such that when n > N, sqrt( 1/(n^2 + 1)^2 + 1/n ) < epsilon. This involves manipulating inequalities, and in this case, we can make the expression smaller by making both a and b smaller than 1.
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demonelite123
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x_n = (n^2 / (n^2+1) , 1/sqrt(n)). prove that this sequence converges and find the limit.

so as n approaches infinite it is clear that x_n approaches (1,0). so using the definition of a convergent sequence, i pick an epsilon > 0 and i have to find some N such that when n>N, sqrt( 1/(n^2 + 1)^2 + 1/n ) < epsilon.

i tried fiddling with the inequality for a while but i can't seem to get it into a form where i can flip the inequality around and say n > (some function of epsilon). i tried combining the fraction into [(n^2+1)^2 + n] / [n(n^2+1)^2] and expanding it but i seem to be getting nowhere. can someone maybe give me a few hints or pushes in the right direction? thanks.

these type of proofs seem to be mostly exercises in manipulating inequalities which i am still inexperienced at.
 
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If you are trying to make something like

[tex]\sqrt{a^2 + b^2}[/tex]

small by making positive numbers a and b small, think about getting each of a and b less than 1. Then you will know a2 < a. Then if you let m = max{a,b}, how big can the square root expression be, and can you make it small by taking a and b additionally smaller than one?
 

What is a convergent sequence?

A convergent sequence is a sequence of numbers that has a limit, meaning that the terms of the sequence get closer and closer to a single value as the sequence progresses.

What does it mean for a sequence to converge?

When a sequence converges, it means that the terms of the sequence get closer and closer to a single value as the sequence progresses. This value is known as the limit of the sequence.

How do you prove that a sequence converges?

To prove that a sequence converges, you must show that as the sequence progresses, the terms of the sequence get closer and closer to a single value, known as the limit. This can be done using various methods, such as the epsilon-delta definition or the monotone convergence theorem.

What is the difference between a convergent and a divergent sequence?

A convergent sequence has a limit, meaning that the terms of the sequence get closer and closer to a single value as the sequence progresses. A divergent sequence, on the other hand, does not have a limit and the terms of the sequence do not approach a single value as the sequence progresses.

Can a sequence converge to more than one limit?

No, a sequence can only have one limit. If the terms of a sequence approach more than one value as the sequence progresses, then the sequence is considered to be divergent.

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