Expert: Paul Klarreich - 1/24/2006

Question
I've decided to tackle a "method of image" problem in electrostatics involving two conducting spheres. I've come across an infinite series I don't recognize, and I was wondering if you could help.

I'm calculating the distance of the image charges and using "L" as the distance between the center of the two identical spheres and "R" as the radius of the spheres, I get the following.

L(0) = L
L(1) = L - (R^2/L)
L(2) = L - (R^2/L) - [R^2/(L - R^2/L)]
L(3) = L - (R^2/L) - [R^2/(L - R^2/L)] - {R^2/[L - (R^2/L) - [R^2/(L - R^2/L)]]}

Or in other words

L(0) = L
L(1) = L(0) - R^2/L(0)
L(2) = L(1) - R^2/L(1)
L(3) = L(2) - R^2/L(2)
.
.
.
L(n) = L(n-1) - R^2/L(n-1)

I need to sum up these terms, but I don't know how to recognize this infinite series. Can you help?

Thanks,

Trent Brunson

Answer
Hi, Trent,
I'm sorry but this doesn't seem familiar.  Actually your second set of definitions:

L(0) = L
L(1) = L(0) - R^2/L(0)
L(2) = L(1) - R^2/L(1)
L(3) = L(2) - R^2/L(2)
.
L(n) = L(n-1) - R^2/L(n-1)

puts this in the realm of what are called Recurrence Relations.  I am not an expert in that area, but I know there are methods for handling Linear RR's, which, unfortunately, yours is not.

I'll look around and if I find out anything, I'll pass it along.

Paul