11.md

title: Problem Set 11

(14.1)

{:.question} Do a Taylor expansion of equation (14.6) around V = 0.

Equation 14.6 states

E \approx 2 E_F - 2 E_C e^{-2/(N_F V)}

The Taylor expansion of e^x about zero is

e^x = \sum_{n = 0}^\infty \frac{x^n}{n!}

so

E \approx 2 E_F - 2 E_C \sum_{n = 0}^\infty \frac{1}{n!} \left( \frac{-2}{N_F V} \right)^n

But this is an expansion around V = \infty.