Hello, these expansions are the reflect that for example 1/49 = 0.02040816326530612244897959... we clearly see powers of 2, but in general we have this Maple procedure : dec:=proc(p) local n, p1, p2, p3, p4, ex; ex := trunc(log10(p) + 1); p2 := trunc(10^ex/p); p3 := trunc(10^ex) - p*trunc(10^ex/p); p4 := ex*n; print(1/p, "can be expanded with this serie"); return 1/p = Sum(p2*p3^(n - 1)/10^p4, n = 1 .. infinity) end; I did that a long time ago ! another example with 1/997. 0.00100300902708124373119358074222668004012036108324974924774322968906720160\ 481444332998996990972918756268806419257773319959879638916750250752256770
dec(997); 1/997, "can be expanded with this serie"
infinity ----- (n - 1) \ 3 1/997 = ) -------- / (3 n) ----- 10 n = 1 in that case where we have 1/999^2 then it is the fact that 1/(1-x)^2 is 1 -------- 2 (1 - x)
series(%,x,23); 2 3 4 5 6 7 8 9 10 1 + 2 x + 3 x + 4 x + 5 x + 6 x + 7 x + 8 x + 9 x + 10 x + 11 x + 12
11 12 13 14 15 16 17 18 19 x + 13 x + 14 x + 15 x + 16 x + 17 x + 18 x + 19 x + 20 x 20 21 22 23 + 21 x + 22 x + 23 x + O(x ) The expansion in base 1000 gives the expansion. Best regards, Simon Plouffe