An example from the last lecture
Find the limit of the sequence
sin3
an =
(n +
(2n+1)π
2
1) sin n1
if it exists.
A. 1
B. 0
C. -1
D. does not exist
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Series
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What is a series?
Definition
Given an ordered list of numbers {a1 , a2 , a3 , · · · }, the sum
a1 + a2 + a3 + · · · =
∞
X
ak
k=1
is called an infinite series or a series for short.
Examples:
1. Geometric series:
2
3
r +r +r =
∞
X
rk.
k=1
2. Harmonic series:
∞
1+
X1
1 1
+ + ··· =
.
2 3
k
k=1
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Do not confuse sequences and series!
A sequence is an ordered list of numbers, such as
1 1 1
1, , , , · · ·
2 3 4
whereas a series is a sum of numbers
1 1
1+ + + · · · .
2 3
A sequence {an } may have aPlimit, but this should not be confused
with the value of the series ∞
k=1 ak . In the above example,
lim an = 0, but we will see later that
n→∞
Math 105 (Section 204)
∞
X
k=1
Series
∞
X
1
ak =
= ∞.
k
k=1
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How does one evaluate a series?
Series as a limit of partial sums
Given a sequence {ak : k ≥ 1}, define the nth partial sum of the sequence
as
Sn = a1 + a2 + · · · + an .
If the sequence of partial sums {Sn : n ≥ 1} has a limit S, i.e.,
lim Sn = S,
n→∞
then we say that the series
P∞
k=1 ak
∞
X
converges and its value is S:
ak = S.
k=1
If the sequence {Sn : n ≥ 1} does not have a limit, we say that the series
diverges.
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An example
Given
ak =
√
k +1−
√
k,
determine P
whether the sequence {ak : k ≥ 1} converges and also whether
the series ∞
k=1 ak converges.
A. the sequence and series both converge
B. the sequence and series both diverge
C. the sequence converges but the series diverges
D. the sequence diverges but the series converges
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Geometric sequence and series
Definition
A sequence is said to be a geometric sequence if each term is a fixed
constant r times the previous term. The constant r is called the common
ratio. Thus a geometric sequence looks like
ak = ar k ,
k ≥ 0.
The corresponding series
∞
X
ak = a
k=1
∞
X
rk
k=1
is called a geometric series.
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Convergence of geometric sequences and series
The sequence ak = ar k converges if and only if −1 < r ≤ 1.


if − 1 < r < 1,
0
k
lim ar = a
if r = 1,

k→∞

DNE if r ≤ −1 or r > 1.
I
Discussion why.
The series
P∞
k=0 ak
∞
X
=
P∞
k=0 ar
k
converges if and only if −1 < r < 1.
ar k = a + ar + ar 2 + · · · =
k=0
a
1−r
if |r | < 1.
The series diverges for |r | ≥ 1.
I
Discussion why.
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Series
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