http://www.ck12.org Chapter 8. Infinite Series
To determine the limit, we look at the trend or behavior of the graph of sequence asngets larger or travels out to
positive infinity. This means we look at the points of sequence that correspond to the far right end of the horizontal
axis in the figure above. We see that the points of the sequence are getting closer to the horizontal axis,y=0. Thus,
the limit of the sequence{ 2 n^1 + 1 }is 0 asntends to infinity. We write: limn→+∞ 2 n^1 + 1 = 0.
Here is the precise definition of the limit of a sequence.
Limit of a Sequence
Thelimit of a sequenceanis the numberLif for eachε>0, there exists an integerNsuch that|an−L|<εfor all
n>N.
Recall that|an−L|<εmeans the values ofansuch thatL−ε<an<L+ε.
What does the definition of the limit of a sequence mean? Here is another example.
Example 5
Look at the figure below.