Reciprocal of 98

Theorem
The decimal expansion of the reciprocal of $98$ starts with the powers of $2$:
 * $\dfrac 1 {98} = 0 \cdotp 0 \dot 1020 \, 40816 \, 32653 \, 06122 \, 44897 \, 95918 \, 36734 \, 69387 \, 75 \dot 5$

Proof
Performing the calculation using long division:

0.01020408163265306122448979591836734693877551... -- 98)1.00000000000000000000000000000000000000000000000     98       196     196    490    294    686     --       ---     ---    ---    ---    ---      200      640     440    900    460    540      196      588     392    882    392    490      ---      ---     ---    ---    ---    ---        400     520     480    180    680    500        392     490     392     98    588    490        ---     ---     ---    ---    ---    ---          800    300     880    820    920    100          784    294     784    784    882     98          ---    ---     ---    ---    ---    ---           160     600    960    360    380   ...            98     588    882    294    294           ---     ---    ---    ---    ---            620     120    780    660    860            588      98    686    588    784            ---     ---    ---    ---    ---             320     220    940    720    760             294     196    882    686    686             ---     ---    ---    ---    ---              260     240    580    340    740              196     196    490    294    686

We have that:
 * $0 \cdotp 01 + 0 \cdotp 0002 + 0 \cdotp 000004 + 0 \cdotp 00000008 + \cdots$

is nothing else but:
 * $\dfrac 1 2 \paren {\dfrac 1 {50} + \paren {\dfrac 1 {50} }^2 + \paren {\dfrac 1 {50} }^3 + \paren {\dfrac 1 {50} }^4 + \cdots} = \displaystyle \dfrac 1 2 \sum_{k \mathop \ge 1} \paren {\dfrac 1 {50} }^k$

Hence: