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 ) n 2 n + 1 = 1 - 1 3 + 1 5 - 1 7 + ,

Doc 30
0.1711
0.1711
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 31
0.1711
0.1711
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 32
0.1683
0.1683
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 33
0.1672
0.1672
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Genetic_drift.html
1 2 1 2 1 2 1 2 = 1 16 .

Doc 34
0.1656
0.6624
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Egyptian_Mathematical_Leather_Roll.html
1 50 + 1 30 + 1 150 + 1 400 = 1 16
1 50 + 1 30 + 1 150 + 1 400 = 1 16
1 25 + 1 15 + 1 75 + 1 200 = 1 8
1 25 + 1 15 + 1 75 + 1 200 = 1 8

Doc 35
0.1649
0.1649
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 + 1 37986 +

Doc 36
0.1631
0.1631
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Logarithmic_growth.html
1 + 1 2 + 1 3 + 1 4 + 1 5 +

Doc 37
0.1597
0.1597
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Generalized_continued_fraction.html
π = 4 1 + 1 2 3 + 2 2 5 + 3 2 7 + = 4 - 1 + 1 6 - 1 34 + 16 3145 - 4 4551 + 1 6601 - 1 38341 + -

Doc 38
0.1585
0.1585
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Net_run_rate.html
254 147.333 + 199 147.333 + 225 147.333 - 253 150 - 110 150 - 103 150 .

Doc 39
0.1557
0.1557
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Expected_value.html
i = 1 x i p i = c ( 1 - 1 2 + 1 3 - 1 4 + )

Doc 40
0.1538
0.1538
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Random_permutation_statistics.html
n ! ( 1 - 1 1 ! + 1 2 ! - 1 3 ! + ± 1 n ! ) = n ! k = 0 n ( - 1 ) k k !

Doc 41
0.1534
0.1534
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000016/Articles/Cours_d'Analyse.html
1 4 , 1 3 , 1 6 , 1 5 , 1 8 , 1 7 ,

Doc 42
0.1524
0.1524
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 43
0.1502
0.2684
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Taylor_series.html
cos x = 1 - x 2 2 ! + x 4 4 ! -
- x - 1 2 x 2 - 1 3 x 3 - 1 4 x 4 -

Doc 44
0.1479
0.1479
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000011/Articles/Rhind_Mathematical_Papyrus_2::n_table.html
2 n = 1 n + 1 2 n + 1 3 n + 1 6 n

Doc 45
0.1448
0.1448
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 46
0.1382
0.1382
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/2_(number).html
k = 0 1 2 k = 1 + 1 2 + 1 4 + 1 8 + 1 16 + = 2.

Doc 47
0.1371
0.1371
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Summation_of_Grandi's_series.html
1 - x + x 2 2 ! - x 3 3 ! + x 4 4 ! - = e - x

Doc 48
0.1358
0.1358
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 49
0.1295
0.1295
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Smith–Volterra–Cantor_set.html
n = 0 2 n 2 2 n + 2 = 1 4 + 1 8 + 1 16 + = 1 2

Doc 50
0.1238
0.1238
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Bernoulli_trial.html
q = 1 - p = 1 - 1 2 = 1 2

Doc 51
0.1190
0.1190
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 + 1 4 +

Doc 52
0.1164
0.1164
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Divergent_series.html
1 + 1 2 + 1 3 + 1 4 + 1 5 + = n = 1 1 n .

Doc 53
0.1154
0.1154
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/William_Brouncker,_2nd_Viscount_Brouncker.html
1 1 + 1 2 2 + 3 2 2 = 13 15 = 1 - 1 3 + 1 5 .

Doc 54
0.1117
0.1117
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Gauss's_continued_fraction.html
π 4 = 1 1 + 1 2 2 + 3 2 2 + 5 2 2 + = 1 - 1 3 + 1 5 - 1 7 + -

Doc 55
0.0964
0.0964
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/1_−_2_+_3_−_4_+_⋯.html
1 2 a 0 - 1 4 Δ a 0 + 1 8 Δ 2 a 0 - = 1 2 - 1 4 .