Sequences & Series: Limit to Determine Convergence/Divergence

[arctic_girl4]

How do you know if a sequence converges or diverges based on taking the limit?

here's an example
f:= 3^n/n^3;

if i take the limit the sequence goes to infinity.

does it diverge because the limit is not zero or can the limit be something other than zero and it still converge?

 

[HallsofIvy]

You are talking about a sequences and not series? You titled this "sequences and series". There is a theorem that says that if a sequence does not converge to 0, then the series (infinite sum) of those numbers cannot converge but certainly if a sequence converges to any number then it converges!

However in this case the sequence "goes to" infinity, as you say, and infinity is not a number. The sequence diverges. Many textbooks would say this sequence "diverges to infinity".