Calculating Madelung Constant Using Evjen Method

[Petar Mali]

Homework Statement

Applying the method of Evjen calculate Madelungovu constant of infinitely long series of alternately opposite charged ions. Show that summarize by the Evjen cells gives the value Madelung constant A_n whose deviation from actual value A is less than \frac{1}{n^2}

Homework Equations

Madelung constant Madelung constant for the infinite number of ions alternately changing signs

A=2(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...)=2ln2

The Attempt at a Solution

Evjen method is method in which we look in neutral structures. So I think that first neutral structure is one whole ion and two neighbours cut in half. And second neutral structure is all other ions. Is it than Madelung constant for first neutral structure

(A)_I=2\cdot 0,5=1

and for second neutral structure

(A)_{II}=2(0,5-\frac{1}{2}+\frac{1}{3}-...)

And

A_n=(A)_I+(A)_{II}

But where I have n in here?

Homework Statement

Homework Equations

The Attempt at a Solution

 

[Petar Mali]

Any idea?