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MATH 1572H Spring 2013 Worksheet 7 Topics:

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MATH 1572H Spring 2013 Worksheet 7 Topics:
MATH 1572H
Spring 2013
Worksheet 7
Topics: absolute and conditional convergence; power series.
Definition. A series
bn is called absolutely convergent if the series
n=1
∞
P
If the series
∞
P
bn converges, while
n=1
∞
P
∞
P
|bn | is convergent.
n=1
|bn | diverges, then it is called conditionally convergent.
n=1
Fact. If a series is absolutely convergent, then it is convergent. This statement may sound a bit
tautological and even silly, but actually it is not totally trivial and requires a proof (which is not
hard though; see p.467).
∞
P
Definition. A series of the form
(−1)n+1 an = a1 − a2 + a3 − a4 + . . . where an ’s are positive
n=1
numbers, is called alternating. There is a simple convergence test for alternating series, which is
due to Leibniz1 :
if a sequence of positive numbers {an } is monotonically decreasing and lim an = 0,
∞
P
then the alternating series
(−1)n+1 an = a1 − a2 + a3 − a4 + . . . converges.
n=1
Notice that this test does not say anything about divergence of an alternating series.
1. For each of the given series, determine if it is absolutely convergent, conditionally convergent
or divergent.
a)
1
3
b)
1
1·3
c)
1
6
−
−
∞
P
n=10
d)
∞
P
n=1
1
+
1
9
1
2·32
−
+
1
12
1
3·33
1
+ · · · + (−1)n+1 3n
+ ...
−
1
4·34
+ · · · + (−1)n+1 n·31 n + . . .
(−1)n
ln n
cos n
n2 +10
Gottfried Wilhelm von Leibniz (1646-1716) was a prominent German mathematician, scientist and
philosopher.
e)
∞
P
n+1
(−1)n+1 2n−1
n=1
2. Prove that the series
3
8
n2 − 1
+ 1−
+ ··· + 1 −
+ ...
1+ 1−
4
9
n2
converges.
Will the series converge if we remove all the parentheses from this formula?
3. For each of the given series, find its interval of convergence.
a)
∞
P
n
x
(−1)n 3n+1
n=1
b)
∞
P
n=1
c)
∞
P
n=1
d)
∞
P
(−3)n n
√ x
n
3n 2n
n! x
√
2013
n xn
n=1
2
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