How can I tell if a sequence converges?

1 Answer
Wataru · Christopher P. · Vinicius M. G. Silveira
Nov 5, 2014

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.

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