asymptotics for sums of uniform random variables Let  U(1) , U(2) , ...   be independent uniform random variables in  [0,1] .
For  x > 0  define  N(x) by
N(x)  =  min { n : U(1) + U(2) + ... + U(n) >  x  } .
Finally, let  
E(x)  =  expected value of the random variable  N(x) .
It is known that   E(1) = e = 2.718...  [e.g. Ross , First course 
in probability; 4th ed. , p. 339 ] .
This clearly gives the upper bound E(x) < x*e  for  x >> 0 .
Questions: a) Is there an asymptotic formula for E(x)?
       b) Does the limit of E(x)/x  exist as x tends to infinity?


Intuitively, the answer to (b) must be "Yes: $2$"
Experimentally $E(100) \approx 200.67$ so I would not be surprised if $E(x) \approx 2x+ \frac23$ for large $x$
To give a handwaving justification, if $x$ is large enough that the starting distribution is irrelevant, then the expected value for the sum immediately after $x$ is exceeded is $x+\frac13$ and so the expected number of terms to reach this sum is double that (as the expected value of each term is $\frac12$). 
