Calculus 2nd Edition Rogawski

SECTION 11.1 Sequences 547
The limit laws we have used for functions also apply to sequences and are proved in
a similar fashion.
THEOREM 2 Limit Laws for Sequences
sequences with
Assume that {a n } and {b n } are convergent
Then:
lim a n = L,
n→∞
lim b n = M
n→∞
(i) lim (a n ± b n ) = lim a n ± lim b n = L ± M
n→∞ n→∞ n→∞
( )( )
(ii) lim a nb n = lim a n lim b n = LM
n→∞ n→∞ n→∞
(iii)
(iv)
a n
lim =
n→∞ b n
lim a n
n→∞
lim b = L
n M
n→∞
lim ca n = c lim a n = cL
n→∞ n→∞
if M ̸= 0
for any constant c
REMINDER n! (n-factorial) is the
number
n! =n(n − 1)(n − 2) ···2 · 1
For example, 4! =4 · 3 · 2 · 1 = 24.
20
10
y
5 10 15
FIGURE 10 Graph of a n = 5n
n
THEOREM 3 Squeeze Theorem for Sequences
such that for some number M,
Then lim
n→∞ a n = L.
Let {a n }, {b n }, {c n } be sequences
b n ≤ a n ≤ c n for n>M and lim
n→∞ b n = lim
n→∞ c n = L
EXAMPLE 7 Show that if lim
n→∞ |a n|=0, then lim
n→∞ a n = 0.
Solution We have
−|a n | ≤ a n ≤ |a n |
By hypothesis, lim |a n|=0, and thus also lim −|a n|=− lim |a n|=0. Therefore,
n→∞ n→∞ n→∞
we can apply the Squeeze Theorem to conclude that lim a n = 0.
n→∞
EXAMPLE 8 Geometric Sequences with r