D(n) denotes the number of digits in the positive integer N.
Evaluate:
∞
Σ {(D(n))!}-1
n=1
(In reply to
Solution by Brian Smith)
Two summations:
clearvars,clc
tot=0;
for k=1:50
term= (9*10^(k-1))/factorial(k);
tot=tot+term;
fprintf('%3d %15.10f %16.10f \n',k,term,tot)
end
tot=0;
for k=1:50
term= (10^(k))/factorial(k);
tot=tot+term;
fprintf('%3d %15.10f %16.10f \n',k,term,tot)
end
The first loop above sums the terms 9*10^(k-1)/k! and eventually the sums come to the total, 19822.9192153260, disagreeing with Brian's answer, which amounts to approximately 24472.7397720074, based on
>> e=exp(1)
e =
2.71828182845905
>> (10/9)*(e^10-1)
ans =
24472.7397720074
The second part verifies the legitimacy of the summation technique by summing 10^k/k!, then manually multiplying by 10/9. The results are shown at bottom: 24472.7397720074.
The problem has to be in factoring out the 10/9. IN fact it is 9/10 that would be factored out. That will make it agree with the computer total.
It seems that (9/10)*(e^10-1) is the answer.
k kth term sum so far
1 9.0000000000 9.0000000000
2 45.0000000000 54.0000000000
3 150.0000000000 204.0000000000
4 375.0000000000 579.0000000000
5 750.0000000000 1329.0000000000
6 1250.0000000000 2579.0000000000
7 1785.7142857143 4364.7142857143
8 2232.1428571429 6596.8571428571
9 2480.1587301587 9077.0158730159
10 2480.1587301587 11557.1746031746
11 2254.6897546898 13811.8643578644
12 1878.9081289081 15690.7724867725
13 1445.3139453139 17136.0864320864
14 1032.3671037957 18168.4535358821
15 688.2447358638 18856.6982717459
16 430.1529599149 19286.8512316608
17 253.0311528911 19539.8823845518
18 140.5728627173 19680.4552472691
19 73.9857172196 19754.4409644887
20 36.9928586098 19791.4338230986
21 17.6156469571 19809.0494700556
22 8.0071122532 19817.0565823088
23 3.4813531536 19820.5379354624
24 1.4505638140 19821.9884992764
25 0.5802255256 19822.5687248020
26 0.2231636637 19822.7918884657
27 0.0826532088 19822.8745416744
28 0.0295190031 19822.9040606776
29 0.0101789666 19822.9142396442
30 0.0033929889 19822.9176326330
31 0.0010945125 19822.9187271456
32 0.0003420352 19822.9190691807
33 0.0001036470 19822.9191728277
34 0.0000304844 19822.9192033122
35 0.0000087098 19822.9192120220
36 0.0000024194 19822.9192144414
37 0.0000006539 19822.9192150953
38 0.0000001721 19822.9192152674
39 0.0000000441 19822.9192153115
40 0.0000000110 19822.9192153225
41 0.0000000027 19822.9192153252
42 0.0000000006 19822.9192153258
43 0.0000000001 19822.9192153260
44 0.0000000000 19822.9192153260
45 0.0000000000 19822.9192153260
46 0.0000000000 19822.9192153260
47 0.0000000000 19822.9192153260
48 0.0000000000 19822.9192153260
49 0.0000000000 19822.9192153260
50 0.0000000000 19822.9192153260
1 10.0000000000 10.0000000000
2 50.0000000000 60.0000000000
3 166.6666666667 226.6666666667
4 416.6666666667 643.3333333333
5 833.3333333333 1476.6666666667
6 1388.8888888889 2865.5555555556
7 1984.1269841270 4849.6825396825
8 2480.1587301587 7329.8412698413
9 2755.7319223986 10085.5731922399
10 2755.7319223986 12841.3051146384
11 2505.2108385442 15346.5159531826
12 2087.6756987868 17434.1916519694
13 1605.9043836822 19040.0960356516
14 1147.0745597730 20187.1705954246
15 764.7163731820 20951.8869686065
16 477.9477332387 21429.8347018453
17 281.1457254346 21710.9804272798
18 156.1920696859 21867.1724969657
19 82.2063524662 21949.3788494319
20 41.1031762331 21990.4820256651
21 19.5729410634 22010.0549667285
22 8.8967913925 22018.9517581209
23 3.8681701706 22022.8199282915
24 1.6117375711 22024.4316658626
25 0.6446950284 22025.0763608911
26 0.2479596263 22025.3243205174
27 0.0918368986 22025.4161574160
28 0.0327988924 22025.4489563084
29 0.0113099629 22025.4602662713
30 0.0037699876 22025.4640362589
31 0.0012161250 22025.4652523840
32 0.0003800391 22025.4656324230
33 0.0001151634 22025.4657475864
34 0.0000338716 22025.4657814580
35 0.0000096776 22025.4657911356
36 0.0000026882 22025.4657938238
37 0.0000007265 22025.4657945503
38 0.0000001912 22025.4657947415
39 0.0000000490 22025.4657947905
40 0.0000000123 22025.4657948028
41 0.0000000030 22025.4657948058
42 0.0000000007 22025.4657948065
43 0.0000000002 22025.4657948067
44 0.0000000000 22025.4657948067
45 0.0000000000 22025.4657948067
46 0.0000000000 22025.4657948067
47 0.0000000000 22025.4657948067
48 0.0000000000 22025.4657948067
49 0.0000000000 22025.4657948067
50 0.0000000000 22025.4657948067
>> e=exp(1)
e =
2.71828182845905
>> (10/9)*(e^10-1)
ans =
24472.7397720074
>> 22025.4657948067*10/9
ans =
24472.7397720074
>>
|
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Posted by Charlie
on 2024-08-27 07:00:07 |
re: Solution
