8.1. Sequences http://www.ck12.org
The figure above shows the graph of the sequence
{ln(n)
n}
. Notice that fromNon, the terms oflnnnare between
L−εandL+ε. In other words, for this value ofε,there is a valueNsuch that all terms ofanare in the interval from
L−εandL+ε. Thus, limn→+∞lnn(n)=0.
Not every sequence has a limit.
Example 6
Here is a graph of the sequence{n+ 1 }.
Consider the sequence{n+ 1 }in the figure above. Asngets larger and goes to infinity, the terms ofan=n+ 1
become larger and larger. The sequence{n+ 1 }does not have a limit. We write limn→+∞(n+ 1 ) = +∞.
Convergence and Divergence
We say that a sequence{an}convergesto a limitLif sequence has a finite limitL. The sequence hasconvergence.
We describe the sequence asconvergent.Likewise, a sequence{an}diverges to a limitLif sequence does not have
a finite limit. The sequence hasdivergenceand we describe the sequence asdivergent.