In this and all subsequent parts
of this problem, you need keep only terms to leading order in N.
That is you may take, for example, 
.
Compute the total energy of N>>1 electrons of mass m in a
square well of width L assuming that each allowed energy
state is occupied by just one electron and that the electrons occupy
the lowest available energy states. From your result, give a formula
for 
the kinetic energy energy per particle of
an electron gas of constant density 
.
Hint: Note that your expression for the energy per particle
should depend only on the density of electrons. Also, you
will find the following sum useful,
