Bob cranked up his sound system. Initially, the volume is set to v>0.
At the end of each second, the control circuitry performs two adjustments:
1.) it resets the volume to the square root of its value
2.) it increases the reset volume by the intial volume, v.
Performing these two operations always increases the volume:
initial volume: v(0) = v
volume after first adjustment:
v(1) = v(0) + sqr(v(0)) = v + sqr(v)
volume after second adjustment:
v(2) = v(1) + sqr(v(1))
= v(0) + sqr(v(0) + sqr(v(0))
= v + sqr(v + sqr(v))
volume after third adjustment:
v(3) = v + sqr(v + sqr(v + sqr(v)))
etc., and obviously v = v(0) < v(1) < v(2) < v(3) < ...
If this process continues indefinately, will the volume converge to a finite value? if so, what value?