Is 1/(n-2) convergent?

[kelsiu]

Suppose there is a sequence Xn=1/(n-2). We know we n tend to infinity the sequence tends to zero. But at n=2 it is equal to infinity. Is this sequence convergent?

There is also a theorem that all convergent sequence are bounded for every n. But the sequence above is not bounded at n=2.

From definition of convergent sequence it seems that only the case that n tends to infinity is concerned, it says nothing about whether it is convergent when n is finite but Xn is not.

In defining whether a sequence is bounded, n is for all values. Is X_n bounded?

 

[JohnZ]

for a sequence usually we only care about it's behavior when n -> infinite.
{1/n-2}, n=1,2,3.. is not well defined. probably we don't say it's divergent.
agree? any thought?

 

[kelsiu]

for a sequence usually we only care about it's behavior when n -> infinite.
{1/n-2}, n=1,2,3.. is not well defined. probably we don't say it's divergent.
agree? any thought?

Thank you. I think if one term of Xn is undefined than the whole thing cannot be called a sequence and it's meaningless to talk about convergence.

 

[stapel]

Suppose there is a sequence Xn=1/(n-2). We know we n tend to infinity the sequence tends to zero. But at n=2 it is equal to infinity.

Actually, the sequence is not defined for n = 2. The sequence is probably only defined for n > 2.