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Section 5.2
137
38. (a) Let
f(x)
=
x+x
2
+X
3
+ .
..
+xn.
The desired sum is
f'(x).
Since
f(x)
is the sum of a geometric
x(1

xn)
x

xn+l
sequence with
a
=
x
and
r
=
x,
we have
f(x)
=
1
=
1
. Thus
x
x
,
[1 
(n
+
1)xn](1

x)

(x

xn+l)(
1)
1
(n
+
1)xn
+
nx
n
+
l
f(x)=
(1x)2
=
(1X)2 .
(b) As
n
+ 00,
Ixl
n
+ 0, so the sum in (a) approaches (1
~
x)2'
(c)
LetS
=
1+3x+5x
2
+7x
3
+
...
be the desired sum and let
SI
=
1+2x+3x
2
+4x
3
+
...
be the
sum found in (b). WehaveSSI
=
x+2x
2
+3x
3
+4x
4
+
..
=
x(1+2x+3x
2
+4x
3
+ .
.
)
=
XSI'
1+x
Thus
S
=
SI
+
XSI
=
(1
+
X)SI
=
(1 _
x)2'
(d) The desired sum is 1
+
3x(1
+
2x
+
3x
2
+
4x
3
+ .
.. )
=
1
+
3XSl,
where
SI
is the sum found
.
(b) S th
h'
1
3x
1
+
x
+
x
2
In
part
.
0
e sum ere IS
+
(1 _
x)2
(1
x)2 .
39
W<
h
2k

1
2k
1
11Th th .
..
th d·fti
.
eave
ak
=
4
k

l
=
4
k

1

4
k

1
=
2
k

2

4
k

l
'
us
e sum
In
question IS
e 1 erence
between the sums of two geometric sequences. The first has
a
=
2~1
=
2,
r
=
!
and the second has
1
1
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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