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?