This problem falls apart fairly nicely--if you see the pattern in the numerator:

1 = 3¹ - 2¹

5 = 3² - 2²

19 = 3³ - 2³ ... etc.

The pattern in the denominator is much more obvious, being just the powers of 6.

So the series is actually the difference between two series:

 

f(x) = =

g(x) = =



These are both simple geometric series whose sums are 1, and .

Thus, the sum of the original series is 1 - =

1

2