Sequences
Sequences
A sequence is a list of numbers in a definite order:
a1 , a2 , a3 , . . . , an , . . .
Per-Olof Persson
persson@berkeley.edu
The term an is called the nth term
The sequence {a1 , a2 , a3 , . . .} is also denoted by
Department of Mathematics
University of California, Berkeley
{an }
Math 1B Calculus
Limits
{an }∞
n=1
or
Limit Laws for Sequences
Definition
A sequence {an } has the limit L and we write
lim an = L
or
n→∞
an → L as n → ∞
Theorem (Limit Laws for Sequences)
If {an } and {bn } are convergent sequences and c is a constant,
then
if for every ε > 0 there is a corresponding integer N such that
if
n>N
then
lim (an + bn ) = lim an + lim bn
n→∞
n→∞
n→∞
then
n→∞
lim c = c
n→∞
n→∞
lim (an bn ) = lim an · lim bn
n→∞
n→∞
an
limn→∞ an
lim
=
if lim bn 6= 0
n→∞ bn
n→∞
lim
b
h n→∞ipn
p
lim an = lim an if p > 0 and an > 0
n→∞
n→∞
an > M
Useful Limit Theorems
Theorem
If limx→∞ f (x) = L and f (n) = an when n is an integer, then
limn→∞ an = L.
Limits
Powers
Theorem (Squeeze Theorem for Sequences)
If an ≤ bn ≤ cn for n ≥ n0 and limn→∞ an = limn→∞ cn = L,
then limn→∞ bn = L.
Theorem
If limn→∞ |an | = 0, then limn→∞ an = 0.
Theorem
If limn→∞ an = L and the function f is continuous at L, then
lim f (an ) = f (L)
n→∞
n→∞
n→∞
Definition
limn→∞ an = ∞ means that for every positive number M there is
an integer N such that
n>N
n→∞
lim can = c lim an
If limn→∞ an exists, the sequence converges (or is convergent).
Otherwise, the sequence diverges (or is divergent).
if
n→∞
lim (an − bn ) = lim an − lim bn
|an − L| < ε
lim
n→∞
1
=0
nr
if r > 0
Exponentials
The sequence {rn } is convergent if −1 < r ≤ 1 and divergent for
all other values of r.
(
0 if − 1 < r < 1
lim rn =
n→∞
1 if r = 1
Increasing, Decreasing, and Bounded Sequences
Definition
A sequence {an } is called increasing if an < an+1 for all n ≥ 1,
i.e., a1 < a2 < a3 < · · · . It is called decreasing if an > an+1 for all
n ≥ 1. It is called monotonic if it is either increasing or decreasing.
Definition
A sequence {an }
is bounded above if there is a number M such that an ≤ M
for all n ≥ 1.
is bounded below if there is a number m such that m ≤ an
for all n ≥ 1.
is a bounded sequence if it is bounded above and below.
Theorem (Monotonic Sequence Theorem)
Every bounded, monotonic sequence is convergent.