Performing the calculation using long division: <pre>    0.02040816326530612244897959183673469387755102...   --------------------------------------------- 49)1.00000000000000000000000000000000000000000000000      98      294     196    441    196    245       --      ---     ---    ---    ---    ---       200     260     240     90    340    250       196     245     196     49    294    245       ---     ---     ---     --    ---    ---         400    150     440    410    460     50         392    147     392    392    441     49         ---    ---     ---    ---    ---     --           80     300    480    180    190     100           49     294    441    147    147      98           --     ---    ---    ---    ---     ---           310      60    390    330    430    ...           294      49    343    294    392           ---      --    ---    ---    ---            160     110    470    360    380            147      98    441    343    343            ---     ---    ---    ---    ---             130     120    290    170    360              98      98    245    147    343              ---    ---    ---    ---    ---              320     220    450    230    270              294     196    441    196    245              ---     ---    ---    ---    --- </pre> Adding up powers of $2$, shifted appropriately to the right: <pre> 02   04     08       16         32           64            128              256                512                 1024                   2048                     4096                       8192                        16384                          32768                            65536                             131072                               262144                                 524288                                  1048576                                    2097152                                      4194304                                        8388608                                         16777216                                           33554432                                             67108864                                              134217728                                                268435456                                                  536870912                                                   1073741824                                                     2147483648                                                       4294967296                                                         8589934592                                                          ........... -------------------------------------------------------------------- 020408163265306122448979591836734693877551020408163265305947144192 --------------------------------------------------------------------                      1 1 1111 11  11212122222112222222322222111 </pre> As can be seen, the decimal expansion of $\dfrac 1 {49}$ matches the sum of the shifted powers of $2$ (to the limits of calculation) and the pattern is apparent. This is what is to be expected, because: :$0 \cdotp 02 + 0 \cdotp 0004 + 0 \cdotp 000008 + 0 \cdotp 00000016 + \cdots$ is nothing else but: :$\dfrac 1 {50} + \paren {\dfrac 1 {50} }^2 + \paren {\dfrac 1 {50} }^3 + \paren {\dfrac 1 {50} }^4 + \cdots = \displaystyle \sum_{k \mathop \ge 1} \paren {\dfrac 1 {50} }^k$ Hence: {{begin-eqn}} {{eqn | l = \sum_{k \mathop \ge 1} \paren {\dfrac 1 {50} }^k       | r = \dfrac {1 / 50} {1 - 1 / 50}       | c = Sum of Infinite Geometric Sequence: Corollary 1 }} {{eqn | r = \dfrac 1 {50 - 1}       | c =  }} {{eqn | r = \dfrac 1 {49}       | c =  }} {{end-eqn}} {{qed}}