How can I tell if a sequence converges?

1 Answer

A sequence #{a_n}# converges if #lim_{n to infty}a_n# exists.

A useful result is that, if the terms of the sequence get arbitrarily close as #n# gets bigger, the sequence is convergent. This is called the Cauchy criterion and the sequence is called a Cauchy sequence.

Written in a more mathematical notation, a sequence is a Cauchy sequence if and only if:

#forall epsilon > 0 exists N : n, m > N => | a_n - a_m | < epsilon#

where #epsilon in RR# and #n, m, N in NN#.


I hope that this was helpful.