Σ(n = 2 to ∞)[1/(n(ln n)²)] = ?

First of all, we can divide the summation as follows:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Invln1.jpg

Then for the latter part, please take a look at the diagram below:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Invln2.jpg

【此圖乃本人自製圖片，未經本人同意勿擅自連結或使用】
From the graph, we can see that the total shaded area (sum of rectangles) is exactly the value of the latter part and, for the unshaded part (white), we can treat each of them as a triangle with base = 1 as shown in the diagram (certainly there's some approximation used).
So to speak, the total unshaded area will be:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Invln3.jpg

And for the total area under the curve, we can evaluate it by means of an improper definite integral as follows:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Invln4.jpg

Finally, we use: Shaded area = Area under the curve - unshaded area to get:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Invln5.jpg

Σ(n = 2 to ∞)[1/(n(ln n)²)] = integral(n=2 to ∞)[1/(n(ln n)²)] dn

=integral(n=2 to ∞)[1/(ln n)²)] d (ln n)

=integral(n=2 to ∞)d (-1/ ln n)

=-1/(ln ∞) - (- 1/ln 2) =1/ (ln 2)= 1.44.

Σ(n = 2 to ∞)[1/(n(ln n)²)] = ?
雖然條式是收歛的﹐不過無真確值的表達式
當n=2,3,4,.....100...951...1000時﹐其值如下
value 代表[1/(i(ln i)²)]
sum代表Σ(i = 2 to n)[1/(i(ln i)²)]
你見到去到1000附近已開始收歛得很慢
用software可得Σ(n = 2 to ∞)[1/(n(ln n)²)] =2.10974






n
value
sum
n
value
sum
n
value
sum

2
1.040684
1.040684
51
0.001268
1.856039
951
2.24E-05
1.963929

3
0.276178
1.316863
52
0.001232
1.857271
952
2.23E-05
1.963951

4
0.130086
1.446949
53
0.001197
1.858468
953
2.23E-05
1.963973

5
0.077211
1.52416
54
0.001164
1.859632
954
2.23E-05
1.963995

6
0.051915
1.576075
55
0.001132
1.860764
955
2.22E-05
1.964018

7
0.037727
1.613802
56
0.001102
1.861866
956
2.22E-05
1.96404

8
0.028908
1.64271
57
0.001073
1.862939
957
2.22E-05
1.964062

9
0.023015
1.665725
58
0.001046
1.863985
958
2.21E-05
1.964084

10
0.018861
1.684586
59
0.001019
1.865004
959
2.21E-05
1.964106

11
0.015811
1.700396
60
0.000994
1.865999
960
2.21E-05
1.964128

12
0.013496
1.713892
61
0.00097
1.866969
961
2.21E-05
1.96415

13
0.011692
1.725584
62
0.000947
1.867915
962
2.2E-05
1.964173

14
0.010256
1.73584
63
0.000925
1.86884
963
2.2E-05
1.964195

15
0.009091
1.744931
64
0.000903
1.869744
964
2.2E-05
1.964216

16
0.00813
1.753061
65
0.000883
1.870626
965
2.19E-05
1.964238

17
0.007328
1.760389
66
0.000863
1.87149
966
2.19E-05
1.96426

18
0.00665
1.767039
67
0.000844
1.872334
967
2.19E-05
1.964282

19
0.006071
1.77311
68
0.000826
1.87316
968
2.19E-05
1.964304

20
0.005571
1.778682
69
0.000808
1.873968
969
2.18E-05
1.964326

21
0.005137
1.783819
70
0.000791
1.87476
970
2.18E-05
1.964348

22
0.004757
1.788576
71
0.000775
1.875535
971
2.18E-05
1.964369

23
0.004422
1.792999
72
0.000759
1.876294
972
2.17E-05
1.964391

24
0.004125
1.797124
73
0.000744
1.877038
973
2.17E-05
1.964413

25
0.003861
1.800985
74
0.000729
1.877768
974
2.17E-05
1.964435

26
0.003623
1.804608
75
0.000715
1.878483
975
2.17E-05
1.964456

27
0.00341
1.808018
76
0.000702
1.879185
976
2.16E-05
1.964478

28
0.003216
1.811234
77
0.000688
1.879873
977
2.16E-05
1.964499

29
0.003041
1.814275
78
0.000675
1.880548
978
2.16E-05
1.964521

30
0.002881
1.817157
79
0.000663
1.881211
979
2.15E-05
1.964543

31
0.002736
1.819892
80
0.000651
1.881862
980
2.15E-05
1.964564

32
0.002602
1.822494
81
0.000639
1.882502
981
2.15E-05
1.964586

33
0.002479
1.824973
82
0.000628
1.88313
982
2.15E-05
1.964607

34
0.002365
1.827338
83
0.000617
1.883747
983
2.14E-05
1.964628

35
0.00226
1.829598
84
0.000606
1.884353
984
2.14E-05
1.96465

36
0.002163
1.831761
85
0.000596
1.884949
985
2.14E-05
1.964671

37
0.002073
1.833834
86
0.000586
1.885535
986
2.13E-05
1.964693

38
0.001989
1.835823
87
0.000576
1.886112
987
2.13E-05
1.964714

39
0.00191
1.837733
88
0.000567
1.886678
988
2.13E-05
1.964735

40
0.001837
1.83957
89
0.000558
1.887236
989
2.13E-05
1.964756

41
0.001769
1.841339
90
0.000549
1.887785
990
2.12E-05
1.964778

42
0.001704
1.843043
91
0.00054
1.888325
991
2.12E-05
1.964799

43
0.001644
1.844687
92
0.000532
1.888856
992
2.12E-05
1.96482

44
0.001587
1.846274
93
0.000523
1.88938
993
2.11E-05
1.964841