Adding Up an Ionic Crystal's Energy

In a line of alternating + and − ions, the energy of one ion is a sum of ±1/n terms. That alternating harmonic series converges, but only conditionally: add the same terms in a different order and you get a different total. Physicists computing crystal energies have to sum in a careful order for exactly this reason.

How it works

Absolute and conditional convergence

In one sentence

$\sum a_n$ converges absolutely if $\sum|a_n|$ converges, and conditionally if $\sum a_n$ converges but $\sum|a_n|$ doesn't. Absolutely convergent series can be rearranged freely; by Riemann's theorem, a conditionally convergent one can be rearranged to any sum.

General formula

{\htmlData{sym=0}{M}} = 2\left(1 - \frac12 + \frac13 - \frac14 + \cdots\right) = 2\ln 2, \qquad \sum \frac{1}{\htmlData{sym=1}{{n}}} = \infty

Conditional convergence is a cancellation act between two infinite piles, positives and negatives. Take from one pile faster and the balance shifts.

What the symbols mean

$M$
Madelung sum
The ion's energy in units of e²/(4πε₀a).
$n$
Distance
How many spacings away a neighbor is.

An ion feels its neighbors at distances $1, 2, 3, \ldots$ with alternating charges, so its energy involves $2\left(1 - \frac12 + \frac13 - \cdots\right)$ .
The alternating series test says this converges (terms alternate, shrink, and go to 0), to $2\ln 2$ .
But the absolute values form the harmonic series, which diverges: the convergence is only conditional.
Rearranging a conditionally convergent series can change its sum. Two positive terms then one negative gives $\frac32\ln 2$ instead of $\ln 2$ .

Learn the math

Alternating SeriesCalc II · Lesson 31, Sequences and Series Absolute and Conditional ConvergenceCalc II · Lesson 32, Sequences and Series

Adding Up an Ionic Crystal's Energy