c Dr Oksana Shatalov, Spring 2016
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10.2: SERIES
A series is a sum of sequence:
∞
X
ak = a1 + a2 + a3 + . . . + an + . . .
k=1
ak is called a general (common) term of the given series. For a given sequence1 {ak }∞
k=1 define the
following:
The sn ’s are called partial sums and they form a sequence {sn }∞
n=1 .
∞
We want to consider the limit of {sn }n=1 :
If
{sn }∞
n=1
is convergent and lim sn = s exists as a real number, then the series
n→∞
n
X
ak is
k=1
convergent. The number s is called the sum of the series.2
∞
X
∞
If {sn }n=1 is divergent then the series
ak is divergent.
k=1
1
k = 1 for convenience, it can be anything
n
X
2
When we write
ak = s we mean that by adding sufficiently many terms of the series we can get as close as
k=1
we like to the number s.
c Dr Oksana Shatalov, Spring 2016
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GEOMETRIC SERIES
a + ar + ar2 + . . . + arn−1 + . . .
(a 6= 0)
Each term is obtained from the preceding one by multiplying it by the common ratio r.
THEOREM 1. The geometric series is convergent if |r| < 1 and its sum is
∞
X
arn−1 =
n=1
If |r| ≥ 1, the geometric series is divergent.
Proof.
∞
X
n=0
arn =
a
.
1−r
c Dr Oksana Shatalov, Spring 2016
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EXAMPLE 2. Determine whether the following series converges or diverges. If it is converges,
find the sum. If it is diverges, explain why.
∞
X
n
2
(a)
5·
7
n=1
(b)
∞
X
(−4)3n
n=0
(c) 1 −
5n−1
3 9 27
+ −
+ ...
2 4
8
c Dr Oksana Shatalov, Spring 2016
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TELESCOPING SUM
Let bn be a given sequence. Consider the following series:
∞
X
(bn − bn+1 )
n=1
EXAMPLE 3. Determine whether the following series converges or diverges. If it is converges,
find the sum. If it is diverges, explain why.
(a)
∞ X
n=1
(b)
∞
X
n=1
(c)
∞
X
n=1
1
1
sin − sin
n
n+1
ln
n+1
n+2
1
n(n + 1)
c Dr Oksana Shatalov, Spring 2016
THEOREM 4. If the series
∞
X
5
an is convergent, then lim an = 0.
n→∞
n=1
REMARK 5. The converse is not necessarily true.
THE TEST FOR DIVERGENCE:
If lim an does not exist or if lim an 6= 0, then the series
n→∞
n→∞
∞
X
an is divergent.
n=1
REMARK 6. If you find that lim an = 0 then the Divergence Test fails and thus another test
n→∞
must be applied.
EXAMPLE 7. Use the test for Divergence to determine whether the series diverges.
(a)
∞
X
n=1
n2
3(n + 1)(n + 2)
c Dr Oksana Shatalov, Spring 2016
(b)
∞
X
cos
n=1
(c)
6
πn
2
∞
X
(−1)n
n=1
n2
THEOREM 8. If
n
X
an and
k=1
c is a constant),
n
X
k=1
(an + bn ), and
k=1
can = c
n
X
k=1
bn are convergent series, then so are the series
n
X
n
X
can (where
k=1
(an − bn ) , and
k=1
k=1
n
X
n
X
an ,
n
n
n
X
X
X
bn ,
an +
(an + bn ) =
k=1
k=1
k=1
n
X
k=1
(an − bn ) =
n
X
k=1
an −
n
X
k=1
bn
c Dr Oksana Shatalov, Spring 2016
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EXAMPLE 9. Determine whether the following series converges or diverges. If it is converges,
find its sum.
∞ X
n=1
1
+5·
n(n + 1)
n 2
7