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learningphysics

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## Main Question or Discussion Point

I'm trying to find out tests with regards to determining if a limit of a sequence exists or not (ie convergence of sequences), since evaluating a particular limit may not always possible.

For example it seems to me that if for a particular sequence a, if

limn->infty a(n+1)/a(n) = 1, then limn->infty a(n) exists.

It also seems like if

limn->infty a(n+1)/a(n) >1, then limn->infty a(n) = infty.

This make sense to me, but I've been searching online for theorems such as these to no avail. Everything I see is with regards to the convergence of series, but not sequences.

Can someone point me to the relevant theorems? Thanks!

For example it seems to me that if for a particular sequence a, if

limn->infty a(n+1)/a(n) = 1, then limn->infty a(n) exists.

It also seems like if

limn->infty a(n+1)/a(n) >1, then limn->infty a(n) = infty.

This make sense to me, but I've been searching online for theorems such as these to no avail. Everything I see is with regards to the convergence of series, but not sequences.

Can someone point me to the relevant theorems? Thanks!