Find the fraction where the decimal expansion is infinite?Find the fraction with integers for the numerator and denominator, where the decimal expansion is 0.11235.....The numerator and denominator must be less than 100.Find the fractionI believe I can use generating functions here to get 1 + x + 2 x 2 + 3 x 3 + 5 x 4 + . . . . . but I do not know how to apply it.

Question

Find the fraction where the decimal expansion is infinite?Find the fraction with integers for the numerator and denominator, where the decimal expansion is   0.11235.....The numerator and denominator must be less than 100.Find the fractionI believe I can use generating functions here to get   1  +  x  +  2      x    2    +  3      x    3    +  5      x    4    +  .  .  .  .  . but I do not know how to apply it.

Kyler Crawford · Accepted Answer

Like you hint, we can observe that the value  0.11235955056  …  =      ∑          k      =      1              ∞            10          −      k            F    k    ,where       F    k   is the   kth Fibonacci number (with the convention that       F    0    =  0,       F    1    =  1), is simply the value of the series  F  (  x  )  :=      ∑          k      =      1              ∞            F    k        x    k    =  x  +      x    2    +  2      x    3    +  3      x    4    +  5      x    5    +  8      x    6    +  13      x    7    +  ⋯at   x  =      1    10  Hint Using the defining recurrence relation      F          k      +      2        =      F          k      +      1        +      F    k  (and the above convention) gives that F(x) satisfies  F  (  x  )  =  x  +  x  F  (  x  )  +      x    2    F  (  x  )  .Rearranging gives that on the open interval of convergence of the series, which turns out to be    (  −  1      /    ϕ  ,  1      /    ϕ  )  ---where   ϕ is the Golden Ratio, and which in particular contains the value       1    10   of interest)---we have  F  (  x  )  =      x          1      −      x      −              x        2              .Thus, the series has value  F      (                  1        10              )    =            (              1        10            )              1      −              (                  1          10                )            −                        (                      1            10                    )                2              =                                                        10              89                                            .Since this is a rational number, its decimal expansion repeats:  0.      1123595505617977528089887640449438202247191    ¯    .

syaoronsangelhwc17 · Accepted Answer

Continued fractions:                              11235          100000                                    =                  2247          20000                                                  =                                                                                                  1                                                                                                                              8                +                                                                                                                                                                  2024                                                                                                                                                                                                              2247                                                                                                                                                                  =                       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                                                                                                                                 2024                                                                                                                                                                                                                                                                              =                                                                                                  1                                                                                                                              8                +                                                                                                                                                                  1                                                                                                                                                                                                              1                        +                                                                                                                                                                                                                                  1                                                                                                                                                                                                                                                                                              9                                +                                                                                                                                                                                                                                                                                                  1                                                                                                                                                                                                                                                                                                                                                                              13                                        +                                                                                                                                                                                                                                                                                                                                                                  2                                                                                                                                                                                                                                                                                                                                                                                                                                                              17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      =                                                                                                  1                                                                                                                              8                +                                                                                                                                                                  1                                                                                                                                                                                                              1                        +                                                                                                                                                                                                                                  1                                                                                                                                                                                                                                                                                              9                                +                                                                                                                                                                                                                                                                                                  1                                                                                                                                                                                                                                                                                                                                                                              13                                        +                                                                                                                                                                                                                                                                                                                                                                  1                                                                                                                                                                                                                                                                                                                                                                                                                                                              8                                                +                                                                                                                                                                                                                                                                                                                                                                                                                                  1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  =        [        0        ;        8        ,        1        ,        9        ,        13        ,        2        ]            The first convergent is 1/8; the second convergent is                                                  1                                                                  8          +          1                      =      1    9    ;the third convergent is                                                  1                                                                  8          +                                                                                                                  1                                                                                                                                                  1                  +                                                                                                                                                                                  1                                                                                                                                                                                                                                  9                                                                                                                                                              =                                                  1                                                                  8          +                                                                                                                  9                                                                                                                                                  10                                                                          =      10    89    ;The fourth convergent can be computed to 131/1166

Find the fraction where the decimal expansion is infinite? Find the fraction with integers for the

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