Show that, if you don’t make too many approximations, the exponential series in equation includes the three-dot diagram in equation 8.18. There will be some leftover terms; show that these vanish in the thermodynamic limit.

Equation:

Limit and Continuity In the ϵ-δ language, lim f(x) = a, (1) x→x 0 is equivalent to the following statement: To each ϵ > 0, there exists a δ > 0 such that |f(x) − a| < ϵ whenever 0 < |x − x |0< δ. The advantage of this deﬁnition is that it does not require any intuition of how big or how small ϵ or δ should be. Example: { 2 f(x) = x x ̸ 2 8 x = 2 √ We claim that lim x→2 f(x) = 4,