8.1. SEQUENCES
451
Activity 8.2.
(a) Recall our earlier work with limits involving infinity in Section 2.8. State clearly
what it means for a continuous function f to have a limit L as x → ∞.
(b) Given that an infinite sequence of real numbers is a function from the integers
to the real numbers, apply the idea from part (a) to explain what you think it
means for a sequence {s n } to have a limit as n → ∞.
(c) Based on your response to (b), decide if the sequence
1+n
2+n
has a limit as
n → ∞. If so, what is the limit? If not, why not?
⊳
In Activities 8.1 and 8.2 we investigated the notion of a sequence {s n } having a limit
as n goes to infinity. If a sequence {s n } has a limit as n goes to infinity, we say that the
sequence converges or is a convergent sequence. If the limit of a convergent sequence is the
number L, we use the same notation as we did for continuous functions and write
lim
n→∞
s n = L.
If a sequence {s n } does not converge then we say that the sequence {s n } diverges.
Convergence of sequences is a major idea in this section and we describe it more formally
as follows.
A sequence {s n } of real numbers converges to a number L if we can make all values
of s k for k ≥ n as close to L as we want by choosing n to be sufficiently large.
Remember, the idea of sequence having a limit as n → ∞ is the same as the idea of
a continuous function having a limit as x → ∞. The only new wrinkle here is that our
sequences are discrete instead of continuous.
We conclude this section with a few more examples in the following activity.
Activity 8.3.
Use graphical and/or algebraic methods to determine whether each of the following
sequences converges or diverges.
(a)
1+2n
3n−2
(b)
5+3 n
10+2 n
(c)
10 n
n!
(where ! is the factorial symbol and n! = n(n − 1)(n − 2) · · · (2)(1) for any
positive integer n (as convention we define 0! to be 1)).
⊳
451
Activity 8.2.
(a) Recall our earlier work with limits involving infinity in Section 2.8. State clearly
what it means for a continuous function f to have a limit L as x → ∞.
(b) Given that an infinite sequence of real numbers is a function from the integers
to the real numbers, apply the idea from part (a) to explain what you think it
means for a sequence {s n } to have a limit as n → ∞.
(c) Based on your response to (b), decide if the sequence
1+n
2+n
has a limit as
n → ∞. If so, what is the limit? If not, why not?
⊳
In Activities 8.1 and 8.2 we investigated the notion of a sequence {s n } having a limit
as n goes to infinity. If a sequence {s n } has a limit as n goes to infinity, we say that the
sequence converges or is a convergent sequence. If the limit of a convergent sequence is the
number L, we use the same notation as we did for continuous functions and write
lim
n→∞
s n = L.
If a sequence {s n } does not converge then we say that the sequence {s n } diverges.
Convergence of sequences is a major idea in this section and we describe it more formally
as follows.
A sequence {s n } of real numbers converges to a number L if we can make all values
of s k for k ≥ n as close to L as we want by choosing n to be sufficiently large.
Remember, the idea of sequence having a limit as n → ∞ is the same as the idea of
a continuous function having a limit as x → ∞. The only new wrinkle here is that our
sequences are discrete instead of continuous.
We conclude this section with a few more examples in the following activity.
Activity 8.3.
Use graphical and/or algebraic methods to determine whether each of the following
sequences converges or diverges.
(a)
1+2n
3n−2
(b)
5+3 n
10+2 n
(c)
10 n
n!
(where ! is the factorial symbol and n! = n(n − 1)(n − 2) · · · (2)(1) for any
positive integer n (as convention we define 0! to be 1)).
⊳