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Math 152-copyright Joe Kahlig, 15a
Section 10.2: Series
Definition: Given a sequence {ai }, we can construct an infinite series or series by adding the
terms of the sequence.
∞
X
ai = a1 + a2 + a3 + ...
i=1
Definition: The nth partial sum of a series, denoted sn , is the sum of the first n-terms.
NOTE: If the index starts at i = 1 then
sn =
n
P
ai = a1 + a2 + ... + an
i=1
s1
s2
s3
s4
= a1
= s1 + a2 = a1 + a2
= s2 + a3 = a1 + a2 + a3
= s3 + a4 = a1 + a2 + a3 + a4
Example: Find the s4 for the series:
∞
P
i=4
How To Shift a Series:
Example: Adjust the series
∞
X
i
10
i=3
Definition: Let
∞
P
1
(i−2)2
1
3
so that the index will now start at i=k.
ai be a series with sn being the nth partial sum of this series. If the sequence
i=1
of partial sums {sn } converges to s, i.e. lim sn = s, then we say that the series
n→∞
s or that the series has a sum of s,
∞
P
i=1
to be divergent.
∞
P
ai converges to
i=1
ai = s. If {sn } does not converge, then the series
∞
P
i=1
ai is said
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Math 152-copyright Joe Kahlig, 15a
Theorem: If the series
∞
P
ai is convergent, then lim ai = 0
i→∞
i=1
Theorem: If lim ai 6= 0 or DNE, then the series
i→∞
∞
P
ai is divergent.
i=1
Example: Which of these series do not have a chance at being convergent?
A)
∞
X
n=1
B)
∞
X
3n + 5
n=1
C)
1
n3
∞
X
7 − 2n
cos(e−n )
n=1
Example: The series
∞
P
ai has a nth partial sum given by sn . Will the series converge or diverge?
i=1
sn =
3n + 5
7 − 2n
Example: Determine if the Harmonic series,
∞
P
1
n , converges or diverges.
n=1
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Math 152-copyright Joe Kahlig, 15a
Example: The geometric series is defined by
∞
X
arn−1 = a + ar + ar2 + ar3 + .... Determine the
n=1
values of r where this series will be convergent.
Theorem: If
P
can = c
P
P
an and
P
an (where c is a constant)
P
(an + bn ) =
P
an +
P
bn
(an − bn ) =
P
an −
P
bn
P
bn are covergent series, then so are the following series
Example: Determine if these series are convergent or divergent. If the series is convergent, then give
the sum of the series.
A) 1 −
4 16 64
+
−
+ ....
3
9
27
Math 152-copyright Joe Kahlig, 15a
B)
∞
X
i−1
1
3
10
i=3
C)
∞
X
7 ∗ 4−n 3n−1
n=0
D)
∞
X
i=1
i
ln
i+1
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Math 152-copyright Joe Kahlig, 15a
E)
∞ X
1
i=3
i−2
−
1
i
Example: Use a geometric series to express 0.14 as a ratio of integers.
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Math 152-copyright Joe Kahlig, 15a
Example: Find the values of x so that
∞
X
(4x − 5)n will converge. Find the sum for those values of x.
n=1