Reciprocal of 19

Theorem

The decimal expansion of the reciprocal of $19$ is as follows:

$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$

This sequence is A021023 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division: 

   0.05263157894736842105...
  ------------------------
19)1.00000000000000000000000
     95    152     76 
     --    ---     --
      50    180     40
      38    171     38
      --    ---     --
      120     90     20
      114     76     19
      ---     --     --
        60    140     100
        57    133      95
        --    ---     ---
         30     70    ...
         19     57
         --     --
         110    130
          95    114
          --    ---
          150    160
          133    152
           --    ---
           170     80
           152     76

$\blacksquare$

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