Calculus- Early Transcendentals, 2021a

340 Sequences and Series
5
4
3
2
1
0
f (n)=1/n
• • • • • • • • • •
0 5 10
5
4
3
2
1
0
f (x)=1/x
.
0 5 10
1
0
−1
f (n)=sin(nπ)
• • • • • • • • •
1 2 3 4 5 6 7 8
1
0
−1
f (x)=sin(xπ)
Figure 9.1: Graphs of sequences and their corresponding real functions.
.
Not surprisingly, the properties of limits of real functions translate into properties of sequences quite
easily. The Properties of Limits theorem becomes:
Theorem 9.5: Properties of Sequences
Suppose that lim
n→∞
a n = L and lim
n→∞
b n = M and k is some constant. Then
lim ka n = k lim a n = kL
n→∞ n→∞
lim (a n + b n )= lim a n + lim b n = L + M
n→∞ n→∞ n→∞
lim (a n − b n )= lim a n − lim b n = L − M
n→∞ n→∞ n→∞
lim (a nb n )= lim a n · lim b n = LM
n→∞ n→∞ n→∞
a n
lim = lim n→∞ a n
= L , if M is not 0
n→∞ b n lim n→∞ b n M
Likewise the Squeeze Theorem becomes:
Theorem 9.6: Squeeze Theorem for Sequences
Suppose that a n ≤ b n ≤ c n for all n > N, forsomeN. Iflim
n→∞
a n = lim
n→∞
c n = L, then lim
n→∞
b n = L.
And a final useful fact:
Theorem 9.7: Absolute Value Sequence
lim |a n| = 0 if and only if lim a n = 0.
n→∞ n→∞
This says simply that the size of |a n | gets close to zero if and only if a n gets close to zero.