What is meant by a divergent sequence?

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Answer 1 · 2015-08-30T09:14:23

A divergent sequence is a sequence that fails to converge to a finite limit.

A sequence $a_0, a_1, a_2,... in RR$ is convergent when there is some $a in RR$ such that $a_n -> a$ as $n -> oo$ .

If a sequence is not convergent, then it is called divergent.

The sequence $a_n = n$ is divergent. $a_n -> oo$ as $n->oo$

The sequence $a_n = (-1)^n$ is divergent - it alternates between $+-1$ , so has no limit.

We can formally define convergence as follows:

The sequence $a_0, a_1, a_2,...$ is convergent with limit $a in RR$ if:

$AA epsilon > 0 EE N in ZZ : AA n >= N, abs(a_n - a) < epsilon$

So a sequence $a_0, a_1, a_2,...$ is divergent if:

$AA a in RR EE epsilon > 0 : AA N in ZZ, EE n >= N : abs(a_n - a) >= epsilon$

That is $a_0, a_1, a_2,...$ fails to converge to any $a in RR$ .