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#.
