Answer: 16/3
Work Shown
The infinite geometric sum exists, i.e. the sum converges, because -1 < r < 1 is true here.

The geometric sequence {4, 1, 1/4, 1/16, 1/64, 1/256, 1/1024, 1/4096, 1/16384, ...} has the infinitely many terms sum to 16/3 = 5.33333.....
Here are a few partial sums:
- 4 + 1 = 5
- 4 + 1 + (1/4) = 5.25
- 4 + 1 + (1/4) + (1/16) = 5.3125
- 4 + 1 + (1/4) + (1/16) + (1/64) = 5.328125
- 4 + 1 + (1/4) + (1/16) + (1/64) + (1/256) = 5.33203125
- 4 + 1 + (1/4) + (1/16) + (1/64) + (1/256) + (1/1024) = 5.3330078125
- 4 + 1 + (1/4) + (1/16) + (1/64) + (1/256) + (1/1024) + (1/4096) = 5.333251953125
- 4 + 1 + (1/4) + (1/16) + (1/64) + (1/256) + (1/1024) + (1/4096) + (1/16384) = 5.33331298828125
The partial sums are slowly approaching 16/3 = 5.333333....
We'll never actually arrive at this exact value because we need infinitely many terms to do so.