What is the Radius of Convergence?
If you take math in your first year of college, they teach you about
infinite series and the radius of convergence. An
infinite series is a polynomial in a variable X with
and infinite number of terms ...
f(X) = K0 + K1 X + K2 X2
+ K3 X3 + ...
In particular, let's consider the series with all of the
Ki are one,
f(X) = 1 + X + X2 + X3 + ...
If we pick X=1/2, we have to add up smaller and smaller
terms 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... forever, with
each number half the size of the previous.
Each number takes you half the remaining distance to two: the sum is said
to converge to two.
Adding up an infinite number of numbers doesn't always give a sensible
answer. Suppose all the Ki are one, but
X=2: we then get the sum 1 + 2 + 4 + 8 + 16 ...
which clearly becomes infinite. (Sometimes we say the sum converges
to infinity, but usually we say it diverges). If we set
X=-2, we get an even worse problem: 1 - 2 + 4 - 8 + 16 ...
If we start on the left and add up the numbers in order, we get
1, then -1, then 3, then -5, then 11 ... we flip-flop back and forth
between larger and larger positive and negative numbers.
If you work at it, you can convince yourself that for X bigger than one,
or for X smaller than minus one, the infinite sum doesn't make sense
(it diverges). For X smaller than one and bigger than minus one, the
sum can be done. The radius of convergence for this function
Four more things we should mention.
- It's called a radius of convergence because mathematicians like
to plug in complex numbers, like X = U + i V, where
i * i = -1. They can show that the series converges
inside a circle U2 + V2 = R2,
and diverges outside the circle.
- If you're clever, you can figure out that our sum (with all the
Ki=1), when it converges, always gives
1/(1-X). An infinite series represents a function.
(Taylor's theorm tells us how to get the series from the function.)
Other infinite series Ki
can be used to make other functions.
- This sum 1/(1-X) does become big when X
gets close to X=1.
On the other hand, there doesn't seem to be anything wrong at
X=-1. The convergence of the infinite series at
X=-1 is spoiled because of a problem far away at
X=1, which happens to be at the same distance from zero!
The radius of convergence is usually the distance to the nearest point
where the function blows up or gets weird.
- There is a simple way to calculate the
radius of convergence of a
series Ki (the ratio test).
The series can't possibly
converge unless the terms eventually get smaller and smaller. If
we insist that |Kn+1 Xn+1| be
smaller than |Kn Xn|, that
can happen only if |X| is smaller than
|Kn/Kn+1|. The radius of
convergence of an infinite series is (basically) the value of
|Kn/Kn+1| for large
Last modified: May 24, 1997
James P. Sethna,
Statistical Mechanics: Entropy, Order Parameters, and Complexity,
now available at
Oxford University Press