Algebra

$2 + \frac{6}{7} +\frac {18}{49} + \frac{54}{343} + 1 + \frac{2}{14} + \frac{3}{98}$

Rearrange as

1 + 2 + 6/7 + 2/14 + 18/49 + 54/343 + 3/98   = 

1 + 2 + (6/7 + 1/7)  +  18/49 + 54/343 + 3/98   = 

1 + 2 + 1  +   (126 + 54) / 343  +   (1.5) / 49   =

4  + 180/343  +  7(1.5) / 343   = 

4 + (180 / 343) + (10.5) /343  =

4 + (1800 + 105)  / 3430  

4 + 1905 /3430

4 + 381/686  =

3125 / 686

The sum of the series is

 over 49 end-fraction

229/49

.