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 ...

Adding up an infinite number of numbers doesn't always give a sensible
answer. Suppose all the **K _{i}** are one, but

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
is one.

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**U**, and diverges outside the circle.^{2}+ V^{2}= R^{2} - If you're clever, you can figure out that our sum (with all the
**K**), when it converges, always gives_{i}=1**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**K**can be used to make other functions._{i} - 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
**K**(the_{i}*ratio test*). The series can't possibly converge unless the terms eventually get smaller and smaller. If we insist that**|K**be smaller than_{n+1}X^{n+1}|**|K**, that can happen only if_{n}X^{n}|**|X|**is smaller than**|K**. The radius of convergence of an infinite series is (basically) the value of_{n}/K_{n+1}|**|K**for large_{n}/K_{n+1}|**n**.

- Elastic Theory has Zero Radius of Convergence.
- What is the radius of convergence?
- Famous Asymptotic Series

Last modified: May 24, 1997

James P. Sethna, sethna@lassp.cornell.edu

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).