Work per particle of a NaCl chain

[guyvsdcsniper]

Homework Statement

Find the work per particle required to assemble such a configuration.
Problem

Relevant Equations

W=qV

The problem states to find the work per particle to assemble the following NaCl chain.
I just want to post my work here to verify I have the correct answer.

My work is attached in the image provided.

 

[ergospherical]

Is this chain supposed to be infinite? (or of length 8?)

 

[guyvsdcsniper]

Is this chain supposed to be infinite? (or of length 8?)

Ah it does say infinite, I missed that in the question.

 

[ergospherical]

Right. Well in that case, focus on a particular ion in the chain (labelled "0" below):
$\dots \ominus_{-3} \oplus_{-2} \ominus_{-1} \oplus_0 \ominus_1 \oplus_{2} \ominus_{3} \dots$

Let the interaction energy between $m$ and $n$ be $U(m,n)$. Assuming each ion to be separated by a distance $a$ from its nearest neighbours, what's the sum of the interaction energies of all the pairs including the ion $n=0$, i.e. $U(0,1) + U(-1,0) + U(0,2) + U(-2,0) + \dots$?

How might you use this to work out the total energy per particle, which is proportional to $\sum\limits_{\substack{m,n \\ m<n}} U(m,n)$? Be careful not to double count.

 

[Steve4Physics]

Edited

If you haven't already sorted this out, here are a couple of hints which should help:
- familiarise yourself with the series expansion of $ln(2)$;
- make your working simpler/neater by defining $A = \frac {q^2}{4 \pi \epsilon_0}$.

 

[guyvsdcsniper]

I figured it out. Thank you both for your help.