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Unformatted text preview: for which the following series converge and when they converge absolutely:
a. b. c. sin nx cos nx
n
sin nx cos nx
1 sin 2nx
This series is just
and so it follows from this earlier example that
n
n
2
converges for every number x and converges conditionally unless x is an integer multiple of .
2
1 n cos nx
n
The series we are considering here can be expressed as
cos n x
n
and from the case considered above we see that the series converges if and only if x is not
an integer multiple of 2. In other words, the series converges if and only if x is not an odd
multiple of .
cos 2 nx
n Hint: Use the identity
cos 2 nx 1 1 cos 2nx
2
2
and show that the series is divergent for every number x.
We see from the identity
cos 2nx
1 2 cos 2 nx
n
n
n
that, unless x is an integer multiple of , the convergence of
1 guarantee that the series
n cos 2nx and the divergence of
n cos 2 nx
n
is divergent. On the other hand, if x is an integer multiple of then the given series reduces to
1 which diverges. Therefore the series
n
cos 2 nx
n
diverges for every number x.
d. cos nx 
n 343 Since
cos nx 
cos 2 nx
n
n
for all n and x we deduce from Part c that the series
cos nx 
n
diverges for every number x.
0 e. cos 3 nx
n Hint: Use the identity
cos 3 nx 3 cos nx 1 cos 3nx.
4
4
If x is an integer multiple of 2 then the series
cos 3 nx
n
is
1
n
which diverges. If 3x is an integer multiple of 2 but x is not then, since
cos nx
n
is convergent and
cos 3nx
n
is divergent, the series
cos 3 nx
3 cos nx 1 cos 3nx
n
n
4
4n
is divergent. If 3x is not an integer multiple of 2 then nor is x and, since both
cos 3nx are convergent, so is
n
cos 3 nx .
n
f. cos nx and
n cos 4 nx
n
Given any number x positive integer n we have
cos 4 nx 3 1 cos 2nx 1 cos 4nx .
n
n
n
8n
2
8
In the event that x is an integer multiple of /2 then the series
cos 4 nx
n
is either
1
1
or
n
2n
and is therefore divergent. In the event that x is not an integer multiple of /2, both of the series
cos 2nx
cos 4nx
and
n
n
are convergent and, since
3
8n
is divergent, the series
cos 4 nx
n
is divergent. We conclude that the series 344 cos 4 nx
n
diverges for every number x.
6. With an eye on the preceding exercise give an example of a convergent series
a 3 is divergent.
n
The series
cos 2n
3
3n a n such that the series is convergent but the series
cos
3 2n
3 3 n is divergent.
7. Find the values of x and for which the binomial series
xn
n
is convergent.
n
If is a nonnegative integer then, since 0 whenever n the series
n
n x converges for
every number x. From now on we assume that fails to be a nonnegative integer. Since
x n 1
n 1
lim
x 
nÝ
 xn 
n
we know that the series
xn
n
is divergent whenever x  1 and is absolutely convergent when x  1. Now we need to consider
the cases x 1.
Now we saw earlier that the series
n
is absolutely convergent when 0. We also saw that the nth term of this series fails to approach
0 as n Ý when
1. Finally we s...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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