Series Expansion for V(r) in Quantum Final | Step-by-Step Solution

[quasar987]

Homework Statement

I am ashamed to ask this, but in my quantum final, there was a little mathematically-oriented subquestion that asked to show that the function

V(r)=-\frac{V_0}{1+e^{(r-R)/a}}

(r in [0,infty)) can be written for r>R as

V_0\sum_{n=1}^{\infty}(-1)^ne^{-n(r-R)/a}

The Attempt at a Solution

 

[matt grime]

You know the series expansion of (1+x)^-1 for |x|<1, right? So use it (and don't tell me that exp{(r-R)/a} >1 for r>R, because I know that).

 

[quasar987]

Yeah ok!

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[HallsofIvy]

Or (really the same thing) the "geometric series"
\sum_{n=0}^\infty ar^n= \frac{a}{1- r}