tangent
Not Supported
x
-
1
-
1
2
-
1
4
-
1
5
-
1
6
-
1
9
-
⋯
=
1
Search
Returned 95 matches (100 formulae, 55 docs)
Lookup 16.844 ms, Re-ranking 0.169 ms
Found 222920 tuple postings, 88494 formulae, 15993 documents
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[ documents ]
[ documents-by-formula ]
Doc 1
1.0000
2.5377
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Goldbach–Euler_theorem.html
x
-
1
-
1
2
-
1
4
-
1
5
-
1
6
-
1
9
-
⋯
=
1
x
-
1
=
1
+
1
2
+
1
4
+
1
5
+
1
6
+
1
9
+
⋯
x
-
1
-
1
2
=
1
+
1
5
+
1
6
+
1
7
+
1
10
+
1
11
+
1
12
+
⋯
x
-
1
=
1
+
1
3
+
1
5
+
1
6
+
1
7
+
1
9
+
1
10
+
1
11
+
⋯
x
=
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
1
7
+
1
8
⋯
1
2
=
1
3
+
1
9
+
1
27
+
1
81
+
⋯
1
=
1
2
+
1
4
+
1
8
+
1
16
+
⋯
Doc 2
0.4919
1.4412
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Riemann_series_theorem.html
1
-
1
2
-
1
4
+
1
3
-
1
6
-
1
8
+
1
5
-
1
10
-
1
12
+
⋯
1
-
1
2
+
1
3
-
1
4
+
⋯
1
2
-
1
4
+
1
6
-
1
8
+
1
10
+
⋯
+
1
2
(
2
k
-
1
)
-
1
2
(
2
k
)
+
⋯
1
+
1
3
+
⋯
+
1
2
a
-
1
-
1
2
-
1
4
-
⋯
-
1
2
b
+
1
2
a
+
1
+
⋯
+
1
4
a
-
1
-
1
2
b
+
2
-
⋯
1
2
k
-
1
-
1
2
(
2
k
-
1
)
-
1
4
k
,
k
=
1
,
2
,
…
.
Doc 3
0.4399
0.7071
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/1::2_−_1::4_+_1::8_−_1::16_+_⋯.html
1
-
1
2
-
1
4
+
1
8
-
1
16
+
⋯
=
1
3
.
1
2
-
1
4
+
1
8
-
1
16
+
⋯
=
1
/
2
1
-
(
-
1
/
2
)
=
1
3
.
Doc 4
0.4110
1.6566
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Harmonic_series_(mathematics).html
1
-
1
2
+
1
3
-
1
4
+
1
5
-
⋯
=
ln
2.
∑
n
=
1
∞
(
-
1
)
n
+
1
n
=
1
-
1
2
+
1
3
-
1
4
+
1
5
-
⋯
∑
n
=
0
∞
(
-
1
)
n
2
n
+
1
=
1
-
1
3
+
1
5
-
1
7
+
⋯
=
π
4
.
∑
n
=
1
∞
1
n
=
1
+
1
2
+
1
3
+
1
4
+
1
5
+
⋯
.
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
1
7
+
1
8
+
1
9
+
⋯
area of
rectangles
=
1
+
1
2
+
1
3
+
1
4
+
1
5
+
⋯
.
Doc 5
0.4092
0.4092
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Special_values_of_L-functions.html
1
-
1
3
+
1
5
-
1
7
+
1
9
-
⋯
=
π
4
,
Doc 6
0.4092
0.4092
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Leibniz_formula_for_π.html
1
-
1
3
+
1
5
-
1
7
+
1
9
-
⋯
=
π
4
.
Doc 7
0.3797
3.4063
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Mathematical_constants_and_functions.html
∑
n
=
1
∞
1
n
(
2
n
n
)
=
1
-
1
2
+
1
4
-
1
5
+
1
7
-
1
8
+
⋯
3
3
4
(
1
-
1
2
2
+
1
4
2
-
1
5
2
+
1
7
2
-
1
8
2
+
1
10
2
±
⋯
)
∑
n
=
1
∞
(
-
1
)
n
+
1
n
!
=
1
1
!
-
1
2
!
+
1
3
!
-
1
4
!
+
1
5
!
-
1
6
!
+
⋯
∑
n
=
1
∞
(
-
1
)
⌊
n
-
1
2
⌋
2
n
+
1
=
1
1
+
1
3
-
1
5
-
1
7
+
1
9
+
1
11
-
⋯
π
4
=
∑
n
=
0
∞
(
-
1
)
n
2
n
+
1
=
1
1
-
1
3
+
1
5
-
1
7
+
1
9
-
⋯
∑
n
=
0
∞
(
-
1
)
n
n
!
=
1
0
!
-
1
1
!
+
1
2
!
-
1
3
!
+
1
4
!
-
1
5
!
+
⋯
∑
k
=
1
∞
(
-
1
)
k
s
k
-
1
=
1
1
-
1
2
+
1
6
-
1
42
+
1
1806
±
⋯
π
2
12
=
∑
n
=
1
∞
(
-
1
)
n
+
1
n
2
=
1
1
2
-
1
2
2
+
1
3
2
-
1
4
2
+
1
5
2
-
⋯
∑
n
=
1
∞
1
n
2
n
=
∑
n
=
1
∞
(
-
1
)
n
+
1
n
=
1
1
-
1
2
+
1
3
-
1
4
+
⋯
π
3
32
=
∑
n
=
1
∞
-
1
n
+
1
(
-
1
+
2
n
)
3
=
1
1
3
-
1
3
3
+
1
5
3
-
1
7
3
+
⋯
∑
n
=
0
∞
(
-
1
)
n
x
2
n
+
1
2
n
+
1
=
1
2
-
1
3
⋅
2
3
+
1
5
⋅
2
5
-
1
7
⋅
2
7
+
⋯
For
x
=
1
/
2
=
1
π
(
1
2
-
1
3
⋅
2
3
+
1
5
⋅
2
5
-
1
7
⋅
2
7
+
⋯
)
∑
n
=
1
∞
1
n
3
=
1
1
3
+
1
2
3
+
1
3
3
+
1
4
3
+
1
5
3
+
⋯
=
Doc 8
0.3675
1.0160
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Series_(mathematics).html
∑
n
=
1
∞
(
-
1
)
n
+
1
n
=
1
-
1
2
+
1
3
-
1
4
+
1
5
-
⋯
1
-
1
2
+
1
3
-
1
4
+
1
5
-
⋯
=
∑
n
=
1
∞
(
-
1
)
n
+
1
1
n
=
ln
(
2
)
.
1
+
1
2
+
1
4
+
1
8
+
⋯
+
1
2
n
+
⋯
.
1
+
1
2
+
1
3
+
1
4
+
1
5
+
⋯
=
∑
n
=
1
∞
1
n
.
Doc 9
0.3527
1.8027
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Convergent_series.html
1
1
-
1
2
+
1
4
-
1
8
+
1
16
-
1
32
+
⋯
=
2
3
.
1
1
-
1
2
+
1
3
-
1
4
+
1
5
⋯
=
ln
(
2
)
1
1
+
1
1
+
1
2
+
1
6
+
1
24
+
1
120
+
⋯
=
e
.
1
1
+
1
3
+
1
6
+
1
10
+
1
15
+
1
21
+
⋯
=
2.
1
1
+
1
2
+
1
4
+
1
8
+
1
16
+
1
32
+
⋯
=
2.
1
1
+
1
1
+
1
2
+
1
3
+
1
5
+
1
8
+
⋯
=
ψ
.
1
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
⋯
→
∞
.
1
1
+
1
4
+
1
9
+
1
16
+
1
25
+
1
36
+
⋯
=
π
2
6
.
Doc 10
0.3309
0.3309
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Natural_logarithm_of_2.html
ln
2
=
1
1
-
1
2
+
1
3
-
1
4
+
1
5
-
⋯
.
Doc 11
0.2949
0.2949
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Fraction_(mathematics).html
2
3
-
1
2
=
4
6
-
3
6
=
1
6
Doc 12
0.2922
0.2922
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Conditional_convergence.html
1
-
1
2
+
1
3
-
1
4
+
1
5
-
⋯
=
∑
n
=
1
∞
(
-
1
)
n
+
1
n
Doc 13
0.2893
0.7125
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/List_of_formulae_involving_π.html
∑
n
=
0
∞
(
(
-
1
)
n
2
n
+
1
)
1
=
1
1
-
1
3
+
1
5
-
1
7
+
1
9
-
⋯
=
arctan
1
=
π
4
∑
n
=
0
∞
(
(
-
1
)
n
2
n
+
1
)
3
=
1
1
3
-
1
3
3
+
1
5
3
-
1
7
3
+
⋯
=
π
3
32
∑
n
=
0
∞
(
(
-
1
)
n
2
n
+
1
)
5
=
1
1
5
-
1
3
5
+
1
5
5
-
1
7
5
+
⋯
=
5
π
5
1536
Doc 14
0.2892
0.7338
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Indian_mathematics.html
π
4
=
1
-
1
3
+
1
5
-
1
7
+
⋯
π
4
=
3
4
+
1
3
3
-
3
-
1
5
3
-
5
+
1
7
3
-
7
-
⋯
cos
x
=
1
-
x
2
2
!
+
x
4
4
!
-
x
6
6
!
+
⋯
Doc 15
0.2873
0.2873
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Van_Wijngaarden_transformation.html
1
-
1
3
+
1
5
-
1
7
+
⋯
=
π
4
=
0.7853981
…
Doc 16
0.2672
0.2672
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Geometric_progression.html
1
2
-
1
4
+
1
8
-
1
16
+
⋯
=
1
/
2
1
-
(
-
1
/
2
)
=
1
3
.
Doc 17
0.2582
0.2582
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Cahen's_constant.html
C
=
∑
(
-
1
)
i
s
i
-
1
=
1
1
-
1
2
+
1
6
-
1
42
+
1
1806
-
⋯
≈
0.64341054629.
Doc 18
0.2569
0.2569
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/1_−_2_+_4_−_8_+_⋯.html
a
0
2
-
Δ
a
0
4
+
Δ
2
a
0
8
-
Δ
3
a
0
16
+
⋯
=
1
2
-
1
4
+
1
8
-
1
16
+
⋯
.
Doc 19
0.2523
0.6969
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Kerala_school_of_astronomy_and_mathematics.html
π
4
=
1
-
1
3
+
1
5
-
1
7
+
…
π
4
=
3
4
+
1
3
3
-
3
-
1
5
3
-
5
+
1
7
3
-
7
-
⋯
cos
x
=
1
-
x
2
2
!
+
x
4
4
!
-
x
6
6
!
+
⋯
Doc 20
0.2500
0.6173
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Pi.html
π
=
4
1
-
4
3
+
4
5
-
4
7
+
4
9
-
4
11
+
4
13
-
⋯
π
=
4
1
-
4
3
+
4
5
-
4
7
+
4
9
-
4
11
+
4
13
⋯
.
π
2
6
=
1
1
2
+
1
2
2
+
1
3
2
+
1
4
2
+
⋯
Doc 21
0.2375
0.6567
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Alternating_series.html
(
1
-
1
2
)
-
1
4
+
(
1
3
-
1
6
)
-
1
8
+
(
1
5
-
1
10
)
-
1
12
+
⋯
ln
(
2
)
=
∑
n
=
1
∞
(
-
1
)
n
+
1
n
=
1
-
1
2
+
1
3
-
1
4
+
⋯
.
η
(
s
)
=
∑
n
=
1
∞
(
-
1
)
n
-
1
n
s
=
1
1
s
-
1
2
s
+
1
3
s
-
1
4
s
+
⋯
Doc 22
0.2371
0.2371
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Biexciton.html
V
=
2
(
1
r
12
-
1
r
1
a
-
1
r
1
b
-
1
r
2
a
-
1
r
2
b
+
1
r
a
b
)
Doc 23
0.2333
0.2333
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000015/Articles/List_of_definite_integrals.html
∫
0
∞
x
e
x
+
1
d
x
=
1
1
2
-
1
2
2
+
1
3
2
-
1
4
2
+
…
=
π
2
12
Doc 24
0.2331
0.2331
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Yuktibhāṣā.html
π
4
=
1
-
1
3
+
1
5
-
1
7
+
⋯
+
(
-
1
)
n
2
n
+
1
+
⋯
Doc 25
0.2331
0.2331
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Madhava_of_Sangamagrama.html
π
4
=
1
-
1
3
+
1
5
-
1
7
+
⋯
+
(
-
1
)
n
2
n
+
1
+
⋯
Doc 26
0.2260
0.2260
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Mass_point_geometry.html
O
1
O
2
B
E
=
B
E
-
B
O
2
-
E
O
1
B
E
=
1
-
B
O
2
B
E
-
E
O
1
B
E
=
1
-
1
2
-
3
13
=
7
26
.
Doc 27
0.2222
0.2222
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Sylvester's_sequence.html
1
=
1
2
+
1
3
+
1
7
+
1
43
+
1
1807
+
⋯
.
Doc 28
0.2173
0.2173
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000010/Articles/File:GeometricSegment.png.html
1
2
+
1
4
+
1
8
+
1
16
+
⋯
=
1
Doc 29
0.2145
0.2145
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Kepler's_laws_of_planetary_motion.html
1
r
min
-
1
p
=
1
p
-
1
r
max
Doc 30
0.2130
0.2130
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/2_(number).html
∑
k
=
0
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2
k
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1
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1
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1
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1
8
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16
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2.
Doc 31
0.2128
0.2128
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Logarithmic_growth.html
1
+
1
2
+
1
3
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1
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1
5
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Doc 32
0.2090
0.2090
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Summation_of_Grandi's_series.html
1
-
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x
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3
3
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Doc 33
0.2058
0.2058
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Generalized_continued_fraction.html
π
=
4
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2
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7
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6
-
1
34
+
16
3145
-
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4551
+
1
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1
38341
+
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Doc 34
0.2053
0.2053
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Dirichlet_eta_function.html
η
(
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n
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1
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Doc 35
0.2005
0.2005
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Expected_value.html
∑
i
=
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x
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(
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+
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Doc 36
0.1981
0.1981
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Gauss's_continued_fraction.html
π
4
=
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1
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+
5
2
2
+
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=
1
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3
+
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5
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1
7
+
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Doc 37
0.1979
0.1979
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/1_−_2_+_3_−_4_+_⋯.html
1
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2
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1
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.
Doc 38
0.1975
0.1975
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000015/Articles/Irrationality_sequence.html
1
2
+
1
3
+
1
7
+
1
43
+
⋯
=
1
,
Doc 39
0.1973
0.1973
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Partial_fraction_decomposition.html
1
18
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2
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3
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Doc 40
0.1931
0.1931
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Bernoulli_trial.html
q
=
1
-
p
=
1
-
1
2
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1
2
Doc 41
0.1892
0.3644
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Taylor_series.html
-
x
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4
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cos
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2
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4
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Doc 42
0.1878
0.1878
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/William_Brouncker,_2nd_Viscount_Brouncker.html
1
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2
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3
2
2
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3
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Doc 43
0.1864
0.3570
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Euler_product.html
π
/
4
=
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3
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2
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1
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1
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Doc 44
0.1853
0.1853
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Prime_zeta_function.html
1
2
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1
3
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1
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7
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1
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+
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.
Doc 45
0.1818
0.1818
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Greedy_algorithm_for_Egyptian_fractions.html
φ
=
1
1
+
1
2
+
1
9
+
1
145
+
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37986
+
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Doc 46
0.1818
0.1818
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Divergent_series.html
1
+
1
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1
3
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1
4
+
1
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+
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∑
n
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1
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1
n
.
Doc 47
0.1807
0.3590
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Egyptian_fraction.html
8
11
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1
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1
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5
12
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1
4
+
1
10
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1
15
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1
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6
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Doc 48
0.1807
0.1807
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Genetic_drift.html
1
2
⋅
1
2
⋅
1
2
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1
2
=
1
16
.
Doc 49
0.1801
0.1801
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Random_permutation_statistics.html
n
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1
n
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n
!
∑
k
=
0
n
(
-
1
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k
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Doc 50
0.1795
0.7179
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Egyptian_Mathematical_Leather_Roll.html
1
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1
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1
200
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1
8
1
25
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1
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+
1
75
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1
200
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Doc 51
0.1771
0.1771
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Smith–Volterra–Cantor_set.html
∑
n
=
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∞
2
n
2
2
n
+
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1
4
+
1
8
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1
16
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=
1
2
Doc 52
0.1766
0.1766
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Divergence_of_the_sum_of_the_reciprocals_of_the_primes.html
∑
n
=
1
∞
1
n
=
1
+
1
2
+
1
3
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4
+
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Doc 53
0.1726
0.1726
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Rhind_Mathematical_Papyrus_2::n_table.html
2
n
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1
n
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1
2
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Doc 54
0.1703
0.1703
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Net_run_rate.html
254
147.333
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199
147.333
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225
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-
253
150
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110
150
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103
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Doc 55
0.1702
0.1702
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Cours_d'Analyse.html
1
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