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Limiting Ratio (Posted on 2011-04-22) Difficulty: 4 of 5
F(n) is a function defined as:

F(n) = Σj=1 to n j6/2j

Evaluate this limit:

Limit F(n)
n → ∞

See The Solution Submitted by K Sengupta    
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Solution No proof but (spoiler) ... | Comment 2 of 3 |

... the sum seems to be 9366.

    5   point 10
   10          Pwr2=2
   20          for J=1 to 200
   30            Term=J*J*J*J*J*J/Pwr2
   40            Tot=Tot+Term
   50            print J,Tot
   60            Pwr2=Pwr2*2
   70          next
  
n       Sum  
1       0.5
2       16.5
3       107.625
4       363.625
5       851.90625
6       1580.90625
7       2500.0390625
8       3524.0390625
9       4562.009765625
10      5538.572265625
11      6403.59228515625
12      7132.59228515625
13      7721.8023681640625
14      8181.3687744140625
15      8528.983062744140625
16      8784.983062744140625
17      8969.13809967041015625
18      9098.88443756103515625
19      9188.6173343658447265625
20      9249.6524906158447265625
21      9290.548964023590087890625
22      9317.580839633941650390625
23      9335.22809278964996337890625
24      9346.61871778964996337890625
25      9353.8946754038333892822265625
26      9358.4978791773319244384765625
27      9361.384386755526065826416015625
28      9363.179568029940128326416015625
29      9364.28751280345022678375244140625
30      9364.96644696034491062164306640625
31      9365.3797230669297277927398681640625
32      9365.6297230669297277927398681640625
33      9365.780069726170040667057037353515625
34      9365.869989177794195711612701416015625
35      9365.92348974887863732874393463134765625
36      9365.95516610090271569788455963134765625
37      9365.9738342175041907466948032379150390625
38      9365.9847879402586841024458408355712890625
39      9365.991188497873736196197569370269775390625
40      9365.994913788172198110260069370269775390625
41      9365.99707388549222741858102381229400634765625
42      9365.99832194681448527262546122074127197265625
43      9365.9990406025896163555444218218326568603515625
44      9365.9994530762775184484780766069889068603515625
45      9365.999689083491290375604876317083835601806640625
46      9365.999823721348008120912709273397922515869140625
47      9365.99990031227950026959661045111715793609619140625
48      9365.99994376406554152936223545111715793609619140625
49      9365.999968351130727128861508390400558710098266601562
50      9365.999982228918534943318263685796409845352172851562
51      9365.999990043240080783704115674481727182865142822265
52      9365.999994433197292753590090796933509409427642822265
53      9365.999996893935697905853032807499403133988380432128
54      9365.999998270329768623732036303408676758408546447753
55      9365.999999038621882579969124549279513303190469741821
56      9365.999999466626451255502061599145235959440469741821
57      9365.999999704605859997263994642935358569957315921783
58      9365.999999836683169308904073346866425708867609500885
59      9365.999999909854731245176240328476069407770410180091
60      9365.999999950322360492763196226917443709680810570716
61      9365.999999972665761216548269141313198815623763948678
62      9365.999999984982348980120520076653178875858429819345
63      9365.999999991761160672760523626667605867623933590948
64      9365.999999995486450971222437689167605867623933590948
65      9365.999999997530684097774228771826621198215434560552
66      9365.999999998650853999922422446489100877897726604714
67      9365.999999999263823840342133800890289663243493123445
68      9365.99999999959879990112061399001874839598258404294
69      9365.999999999781620380594153300359006042441478712134
70      9365.999999999881272959805674480950055116323937909328
71      9365.99999999993552554032253312636148445680106533473
72      9365.999999999965026442044024017211448218667158016614
73      9365.999999999981049576508276628304065086760843428237
74      9365.99999999998974259571653047797912608474213413956
75      9365.999999999994453647798961393408667154558688849874
76      9365.999999999997004010118296119200031503437831048586
77      9365.99999999999838323373455035102978978069077076185
78      9365.999999999999128356709178478879418991778771363954
79      9365.999999999999530511169902114177757362998427703715
80      9365.999999999999747351604399215064558853558601692057
81      9365.999999999999864161722893925692682402122692109092
82      9365.99999999999992702882906610352621515488361764179
83      9365.99999999999996083367135506415075688592072368499
84      9365.999999999999978995353916363885919604614438763179
85      9365.999999999999988744440396325297863509982325500718
86      9365.999999999999993973349355367192560516491058129201
87      9365.999999999999996775593000442715429386251712305135
88      9365.999999999999998276163535513490992366468189931662
89      9365.999999999999999079080114975433289098407391989409
90      9365.999999999999999508374735986950457615337379022957
91      9365.999999999999999737735300319618446472368719267975
92      9365.999999999999999860187717662470457963560665680879
93      9365.999999999999999925517031170910346587269971975727
94      9365.99999999999999996034655436889161900264152149676
95      9365.999999999999999978902882505181619051139190540057
96      9365.999999999999999988782674801955778206032104187985
97      9365.999999999999999994039467200612194585326388630826
98      9365.999999999999999996834692890418239968626295479065
99      9365.999999999999999998320086656226412559593133378985
100     9365.999999999999999999108947561447424365004861944268
101     9365.999999999999999999527643434904149184439990450983
102     9365.999999999999999999749739811275770100287084613153
103     9365.999999999999999999867482458215906969356696483624
104     9365.999999999999999999929867502361164265580421316711
105     9365.99999999999999999996290341049073205378616876151
106     9365.999999999999999999980388008474600401000086589964
107     9365.99999999999999999998963697334844610168636257537
108     9365.999999999999999999994526907390905414434000731004
109     9365.999999999999999999997110889249468481805989362841
110     9365.999999999999999999998475650200751920180019643377
111     9365.999999999999999999999196107679636822456751319334
112     9365.999999999999999999999576252101067491022848766058
113     9365.999999999999999999999776736762223449002190712781
114     9365.999999999999999999999882420849812163143091922162
115     9365.999999999999999999999938105762208468811013761041
116     9365.999999999999999999999967432816174133091080961676
117     9365.999999999999999999999982871336638061523639826287
118     9365.999999999999999999999990995011919005029742909188
119     9365.9999999999999999999999952678091581334560331699
120     9365.999999999999999999999997514213845266727932174392
121     9365.999999999999999999999998694759390001415124232721
122     9365.999999999999999999999999314913338300729511104254
123     9365.999999999999999999999999640555929589609912095079
124     9365.999999999999999999999999811482921088789622995315
125     9365.999999999999999999999999901166022805577915534433
126     9365.99999999999999999999999994820347793537267647582
127     9365.999999999999999999999999972864602475298503425435
128     9365.999999999999999999999999985789299546439560845301
129     9365.999999999999999999999999992560549118404144585935
130     9365.999999999999999999999999996106728313403003800967
131     9365.999999999999999999999999997963242797980760037608
132     9365.999999999999999999999999998934835311150317239432
133     9365.999999999999999999999999999443135677241482515608
134     9365.999999999999999999999999999708968970301940122604
135     9365.999999999999999999999999999847949254790559561824
136     9365.999999999999999999999999999920585609355892480766
137     9365.999999999999999999999999999958535803062142043063
138     9365.999999999999999999999999999978357239164147332084
139     9365.99999999999999999999999999998870674000667340682
140     9365.999999999999999999999999999994108917150419160844
141     9365.999999999999999999999999999996927853613711431454
142     9365.999999999999999999999999999998398372755194503813
143     9365.999999999999999999999999999999165251754324310213
144     9365.999999999999999999999999999999565063526033332482
145     9365.999999999999999999999999999999773444777598804411
146     9365.999999999999999999999999999999882021760074867162
147     9365.999999999999999999999999999999938579838514067935
148     9365.999999999999999999999999999999968032933236089946
149     9365.999999999999999999999999999999983366678937727409
150     9365.999999999999999999999999999999991347511597827344
151     9365.999999999999999999999999999999995500228624094463
152     9365.999999999999999999999999999999997660469536006868
153     9365.999999999999999999999999999999998783933763036779
154     9365.999999999999999999999999999999999368057685336057
155     9365.999999999999999999999999999999999671685015723734
156     9365.999999999999999999999999999999999829470943040272
157     9365.999999999999999999999999999999999911447296395849
158     9365.999999999999999999999999999999999954027056160625
159     9365.999999999999999999999999999999999976138313309196
160     9365.999999999999999999999999999999999987617750328945
161     9365.999999999999999999999999999999999993576099556924
162     9365.999999999999999999999999999999999996668037648727
163     9365.999999999999999999999999999999999998272155726578
164     9365.999999999999999999999999999999999999104194948865
165     9365.999999999999999999999999999999999999535668700334
166     9365.999999999999999999999999999999999999759370381549
167     9365.999999999999999999999999999999999999875325400443
168     9365.999999999999999999999999999999999999935417367267
169     9365.999999999999999999999999999999999999966552517131
170     9365.999999999999999999999999999999999999982681027783
171     9365.999999999999999999999999999999999999991034122459
172     9365.999999999999999999999999999999999999995359374574
173     9365.999999999999999999999999999999999999997598546131
174     9365.999999999999999999999999999999999999998757526938
175     9365.999999999999999999999999999999999999999357289082
176     9365.999999999999999999999999999999999999999667599796
177     9365.999999999999999999999999999999999999999828120244
178     9365.999999999999999999999999999999999999999911139872
179     9365.999999999999999999999999999999999999999954068693
180     9365.999999999999999999999999999999999999999976262704
181     9365.999999999999999999999999999999999999999987734786
182     9365.999999999999999999999999999999999999999993663617
183     9365.999999999999999999999999999999999999999996727113
184     9365.999999999999999999999999999999999999999998309773
185     9365.99999999999999999999999999999999999999999912726
186     9365.99999999999999999999999999999999999999999954944
187     9365.999999999999999999999999999999999999999999767432
188     9365.999999999999999999999999999999999999999999879972
189     9365.999999999999999999999999999999999999999999938061
190     9365.99999999999999999999999999999999999999999996804
191     9365.999999999999999999999999999999999999999999983509
192     9365.99999999999999999999999999999999999999999999149
193     9365.999999999999999999999999999999999999999999995606
194     9365.999999999999999999999999999999999999999999997729
195     9365.999999999999999999999999999999999999999999998824
196     9365.999999999999999999999999999999999999999999999388
197     9365.999999999999999999999999999999999999999999999679
198     9365.999999999999999999999999999999999999999999999829
199     9365.999999999999999999999999999999999999999999999906
200     9365.999999999999999999999999999999999999999999999945

so apparently the answer is 9366.

The first few terms themselves are:

j       term
1       0.5
2       16.0
3       91.125
4       256.0
5       488.28125
6       729.0
7       919.1328125
8       1024.0
9       1037.970703125
10      976.5625
11      865.02001953125
12      729.0
13      589.2100830078125
14      459.56640625
15      347.614288330078125
16      256.0
17      184.15503692626953125
18      129.746337890625
19      89.7328968048095703125
20      61.03515625
21      40.896473407745361328125
22      27.0318756103515625
23      17.64725315570831298828125
24      11.390625
25      7.2759576141834259033203125
26      4.60320377349853515625
27      2.886507578194141387939453125
28      1.7951812744140625
29      1.10794477351009845733642578125
30      0.678934156894683837890625
31      0.4132761065848171710968017578125
32      0.25
33      0.150346659240312874317169189453125
34      0.0899194516241550445556640625
35      0.05350057108444161713123321533203125
36      0.031676352024078369140625
37      0.0186681166014750488102436065673828125
38      0.01095372275449335575103759765625
39      0.006400557615052093751728534698486328125
40      0.0037252902984619140625
41      0.00216009732002930832095444202423095703125
42      0.001248061322257854044437408447265625
43      0.0007186557751310829189606010913848876953125
44      0.00041247368790209293365478515625
45      0.000236007213771927126799710094928741455078125
46      0.0001346378567177453078329563140869140625
47      0.00007659093149214868390117771923542022705078125

  Posted by Charlie on 2011-04-22 15:23:52
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