tangent
Not Supported
x
-
1
-
1
2
-
1
4
-
1
5
-
1
6
-
1
9
-
⋯
=
1
Search
Returned 97 matches (100 formulae, 55 docs)
Lookup 16.844 ms, Re-ranking 1784.429 ms
Found 222920 tuple postings, 88494 formulae, 15993 documents
[ formulas ]
[ documents ]
[ documents-by-formula ]
Doc 1
1.0000
2.2364
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
+
⋯
1
2
=
1
3
+
1
9
+
1
27
+
1
81
+
⋯
x
=
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
1
7
+
1
8
⋯
1
=
1
2
+
1
4
+
1
8
+
1
16
+
⋯
x
-
1
=
1
+
1
3
+
1
5
+
1
6
+
1
7
+
1
9
+
1
10
+
1
11
+
⋯
Doc 2
0.3697
2.9043
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
±
⋯
)
π
4
=
∑
n
=
0
∞
(
-
1
)
n
2
n
+
1
=
1
1
-
1
3
+
1
5
-
1
7
+
1
9
-
⋯
∑
k
=
1
∞
(
-
1
)
k
s
k
-
1
=
1
1
-
1
2
+
1
6
-
1
42
+
1
1806
±
⋯
∑
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
-
⋯
∑
n
=
0
∞
(
-
1
)
n
n
!
=
1
0
!
-
1
1
!
+
1
2
!
-
1
3
!
+
1
4
!
-
1
5
!
+
⋯
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 3
0.3363
1.0261
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
2
k
-
1
-
1
2
(
2
k
-
1
)
-
1
4
k
,
k
=
1
,
2
,
…
.
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
-
⋯
Doc 4
0.3333
1.6102
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
2
+
1
4
+
1
8
+
1
16
+
1
32
+
⋯
=
2.
1
1
+
1
3
+
1
6
+
1
10
+
1
15
+
1
21
+
⋯
=
2.
1
1
+
1
1
+
1
2
+
1
6
+
1
24
+
1
120
+
⋯
=
e
.
1
1
+
1
1
+
1
2
+
1
3
+
1
5
+
1
8
+
⋯
=
ψ
.
1
1
+
1
4
+
1
9
+
1
16
+
1
25
+
1
36
+
⋯
=
π
2
6
.
1
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
⋯
→
∞
.
Doc 5
0.3155
0.3155
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 6
0.3155
0.3155
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 7
0.3003
0.4686
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 8
0.2888
1.1361
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
.
area of
rectangles
=
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
+
⋯
∑
n
=
1
∞
1
n
=
1
+
1
2
+
1
3
+
1
4
+
1
5
+
⋯
.
Doc 9
0.2809
0.2809
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 10
0.2776
0.6196
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 11
0.2470
0.7160
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
4
+
1
8
+
⋯
+
1
2
n
+
⋯
.
1
-
1
2
+
1
3
-
1
4
+
1
5
-
⋯
=
∑
n
=
1
∞
(
-
1
)
n
+
1
1
n
=
ln
(
2
)
.
1
+
1
2
+
1
3
+
1
4
+
1
5
+
⋯
=
∑
n
=
1
∞
1
n
.
Doc 12
0.2420
0.2420
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Fraction_(mathematics).html
2
3
-
1
2
=
4
6
-
3
6
=
1
6
Doc 13
0.2262
0.2262
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 14
0.2260
0.2260
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 15
0.2226
0.2226
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 16
0.2206
0.5912
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Indian_mathematics.html
π
4
=
3
4
+
1
3
3
-
3
-
1
5
3
-
5
+
1
7
3
-
7
-
⋯
π
4
=
1
-
1
3
+
1
5
-
1
7
+
⋯
cos
x
=
1
-
x
2
2
!
+
x
4
4
!
-
x
6
6
!
+
⋯
Doc 17
0.2206
0.5545
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Kerala_school_of_astronomy_and_mathematics.html
π
4
=
3
4
+
1
3
3
-
3
-
1
5
3
-
5
+
1
7
3
-
7
-
⋯
π
4
=
1
-
1
3
+
1
5
-
1
7
+
…
cos
x
=
1
-
x
2
2
!
+
x
4
4
!
-
x
6
6
!
+
⋯
Doc 18
0.2062
0.2062
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 19
0.2032
0.2032
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000010/Articles/File:GeometricSegment.png.html
1
2
+
1
4
+
1
8
+
1
16
+
⋯
=
1
Doc 20
0.2004
0.2004
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 21
0.1963
0.1963
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000015/Articles/Irrationality_sequence.html
1
2
+
1
3
+
1
7
+
1
43
+
⋯
=
1
,
Doc 22
0.1959
0.1959
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Partial_fraction_decomposition.html
1
18
=
1
2
-
1
3
-
1
3
2
.
Doc 23
0.1897
0.1897
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Prime_zeta_function.html
1
2
+
1
3
+
1
5
+
1
7
+
1
11
+
⋯
→
∞
.
Doc 24
0.1887
0.1887
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 25
0.1878
0.5163
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
2
6
=
1
1
2
+
1
2
2
+
1
3
2
+
1
4
2
+
⋯
π
=
4
1
-
4
3
+
4
5
-
4
7
+
4
9
-
4
11
+
4
13
⋯
.
Doc 26
0.1796
0.3378
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Egyptian_fraction.html
8
11
=
1
2
+
1
22
+
1
6
+
1
66
.
5
12
=
1
4
+
1
10
+
1
15
=
1
5
+
1
6
+
1
20
.
Doc 27
0.1780
0.4652
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Alternating_series.html
η
(
s
)
=
∑
n
=
1
∞
(
-
1
)
n
-
1
n
s
=
1
1
s
-
1
2
s
+
1
3
s
-
1
4
s
+
⋯
ln
(
2
)
=
∑
n
=
1
∞
(
-
1
)
n
+
1
n
=
1
-
1
2
+
1
3
-
1
4
+
⋯
.
(
1
-
1
2
)
-
1
4
+
(
1
3
-
1
6
)
-
1
8
+
(
1
5
-
1
10
)
-
1
12
+
⋯
Doc 28
0.1780
0.1780
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Dirichlet_eta_function.html
η
(
s
)
=
∑
n
=
1
∞
(
-
1
)
n
-
1
n
s
=
1
1
s
-
1
2
s
+
1
3
s
-
1
4
s
+
⋯
Doc 29
0.1748
0.3141
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Euler_product.html
∏
p
(
1
-
3
p
3
+
2
p
4
+
1
p
5
-
1
p
6
)
=
0.678234...
π
/
4
=
∑
n
=
0
∞
(
-
1
)
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Doc 30
0.1711
0.1711
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Yuktibhāṣā.html
π
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Doc 31
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testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Madhava_of_Sangamagrama.html
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Doc 32
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testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Geometric_progression.html
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Doc 33
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Doc 34
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testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Egyptian_Mathematical_Leather_Roll.html
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Doc 35
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testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Greedy_algorithm_for_Egyptian_fractions.html
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Doc 36
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testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Logarithmic_growth.html
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testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Generalized_continued_fraction.html
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Doc 39
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Doc 40
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testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Random_permutation_statistics.html
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testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Cours_d'Analyse.html
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Doc 42
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Doc 43
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0.2684
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Doc 44
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Doc 45
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Doc 46
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Doc 47
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Doc 48
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Doc 49
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Doc 53
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