James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

696 Chapter 11 Infinite Sequences and Series
1 Definition A sequence ha n j has the limit L and we write
lim
n l ` an − L or a n l L as n l `
if we can make the terms a n as close to L as we like by taking n sufficiently large.
If lim n l ` a n exists, we say the sequence converges (or is convergent). Otherwise,
we say the sequence diverges (or is divergent).
Figure 3 illustrates Definition 1 by showing the graphs of two sequences that have
the limit L.
a n
L
a n
L
FIGURE 3
Graphs of two
sequences with
lim
n l ` an − L
0 n
0 n
A more precise version of Definition 1 is as follows.
2 Definition A sequence ha n j has the limit L and we write
Compare this definition with
Definition 2.6.7.
lim
n l ` an − L or a n l L as n l `
if for every « . 0 there is a corresponding integer N such that
if n . N then | a n 2 L | , «
Definition 2 is illustrated by Figure 4, in which the terms a 1 , a 2 , a 3 , . . . are plotted on
a number line. No matter how small an interval sL 2 «, L 1 «d is chosen, there exists an
N such that all terms of the sequence from a N11 onward must lie in that interval.
a¡ a£ a aˆ a N+1 a N+2 a˜ aß a∞ a¢ a
FIGURE 4
0 L-∑ L L+∑
Another illustration of Definition 2 is given in Figure 5. The points on the graph of
ha n j must lie between the horizontal lines y − L 1 « and y − L 2 « if n . N. This
picture must be valid no matter how small « is chosen, but usually a smaller « requires
a larger N.
y
L
y=L+∑
y=L-∑
FIGURE 5
0 1 2 3 4
N
n
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