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Other
Quantitative Aptitude
Functions
If a=∑_n=1^∞ ...
Question
If
a
=
∑
∞
n
=
1
(
2
n
(
2
n
−
1
)
!
)
and
b
=
∑
∞
n
=
1
(
2
n
(
2
n
+
1
)
!
)
. Find value of [a] + [3b] where [ ] denotes greatest integer function or integral value of x.
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Solution
a
=
∑
∞
n
=
1
2
n
(
2
n
+
1
)
!
=
e
=
2.71828
b
=
∑
∞
n
=
1
2
n
(
2
n
+
1
)
!
=
1
e
=
0.36787
[
a
]
+
[
3
b
]
=
[
2.71828
]
+
0
[
3
×
0.3678
]
=
2
+
1
=
3
∴
[
a
]
+
[
3
b
]
=
3
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Similar questions
Q.
If
I
=
x
∫
0
[
sin
t
]
d
t
, where
x
∈
(
2
n
π
,
(
2
n
+
1
)
π
)
,
n
∈
N
and
[
⋅
]
denotes the greatest integer function, then the value of
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is
Q.
Let
f
(
x
)
=
x
−
[
x
]
and
g
(
x
)
=
lim
n
→
∞
[
f
(
x
)
]
2
n
−
1
[
f
(
x
)
]
2
n
+
1
,
then the absolute value of
g
(
x
)
is
(where
[
.
]
denotes the greatest integer function)
Q.
The value of
1
1
!
(
n
−
1
)
!
+
1
3
!
(
n
−
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)
!
+
…
+
1
(
n
−
1
)
!
(
1
)
!
, where
n
is even positive integer, is
Q.
Prove that
1
−
2
n
+
2
n
(
2
n
−
1
)
2
!
−
2
n
(
2
n
−
1
)
(
2
n
−
1
)
3
!
+
.
.
.
+
(
−
1
)
n
−
1
2
n
(
n
−
1
)
.
.
.
(
n
+
2
)
(
n
−
1
)
=
(
−
1
)
n
+
1
(
2
n
)
2
(
n
!
)
2
,
where n is a + ive integer.
Q.
lim
x
→
0
−
∑
2
n
+
1
r
=
1
[
x
r
]
+
(
n
+
1
)
1
+
[
x
]
+
|
x
|
+
2
x
(
where
n
ϵ
N
&
[
.
]
denotes the greatest integer function
)
equals
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