Why Read This Book? - Index of

L 1 e
L
L 2 e
. . . . . .
1 2 3 4 N
Figure 6.2 A convergent sequence.
. . . . . .
6.2 Convergence of Sequences 191
You then take that ɛ value and do a quick calculation to determine some positive
integer N (which will depend on ɛ) with the property that the Nth term and all
those after it fall in the ɛ-neighborhood of L. If someone else then gives you an
even smaller ɛ>0, you can still find a threshold value N with the same property,
though it will probably be a higher threshold. If you are given any ɛ>0 and are
able to find some positive integer N with the property that all terms from the Nth
one onward fall in the ɛ-neighborhood of L, then you will have shown that the
sequence converges to L.
Before we give the definition of convergence, note that you have already done
some of this sort of work in the exercises of Section 6.1. In Exercise 6.1.17, you
showed that
If n>1 + 4
� �
�
, then �
n + 3 �
ɛ � − 1�
n − 1 � 1 7
+ , then
2 4ɛ
� �
�
�
n + 3 1�
� − �
2n − 1 2 � 0,
there exists a positive integer N with the property that n ≥ N implies |an − L| 0)(∃N ∈ N)(∀n ∈ N)(n ≥ N → |an − L|