Question

How can I tell if a sequence converges?

Answer 1

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`.

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I hope that this was helpful.