Question 1118782
<br>
S = (2^1-1)/3^1 + (2^2-1)/3^2 + {2^3-1/3^3 + ...<br>
S = 1/3 + 3/9 + 7/27 + 15/81 + ...<br>
If the denominators of the series form a geometric progression but the whole series is not geometric, then multiply that expression for the sum by the common ratio; then subtract the two series.<br><pre>
  3S = 1 + 3/3 + 7/9 + 15/27 + ...
   S =     1/3 + 3/9 + 7/27 + ...
  --------------------------------
  2S = 1 + 2/3 + 4/9 + 8/27 + ...</pre><br>
The expression on the right is a pure geometric series...<br>
  2S = 1/(1-(2/3)) = 1/(1/3) = 3
   S = 3/2 or 1.5