How do you get the limit of an infinite sequence?

Answer 1

There are a lot of useful techniques. If it's a sequence defined recursively, like

an = an-1 / 2

then you can set an = an-1 and solve for that. If it's defined more simply, like

an = 1/2n, you can consider the behaviour of the sequence as n goes to infinity (These are both the same sequence, with limit 0).

I don't think either of these ways are really considered rigorous though, even though they lead you to the correct answer. To formally prove the convergence of a sequence, you can either use the delta-epsilon or the Cauchy method if it is in Rk

In the delta-epsilon method, you just need to show that for any ε > 0, there exists a δ such that if you have two n1, n2 so that |n1 - n2| < δ, d(an1 - an2) < ε.

To show a function fulfills the Cauchy criterion, you need to show that for any ε > 0, you can find an N so that for all n1, n2 > N, d(an1 - an2) < ε, but this is only a valid proof of convergence in Rk