diff --git a/_psets/11.md b/_psets/11.md index 7803f3851263b9f29bdfd05580b0e9b15c057492..c088810cf79fbdb9dd0f7e8628604f23e133436b 100644 --- a/_psets/11.md +++ b/_psets/11.md @@ -8,6 +8,42 @@ title: Problem Set 11 {:.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 $$V = \infty$$. + +If you really want an expansion about $$V = 0$$, note that + +$$ +\frac{d}{dx} e^{x^{-1}} = -x^{-2} e^{x^{-1}} +$$ + +As we keep taking higher derivatives we'll get more and more negative powers of $$x$$, but we'll +never get rid of the $$e^{1/x}$$. So at $$x = 0$$ the latter term dominates, meaning the function +and all its derivatives are zero. Thus the Taylor expansion about zero is identically zero. + +Plugging this into equation 14.6 gives us the rather uninteresting + +$$ +E \approx 2 E_F +$$ + ## (14.2)