Radius of Convergence - Definition

Definition

For a power series ƒ defined as:

where

a is a complex constant, the center of the disk of convergence,
cn is the nth complex coefficient, and
z is a complex variable.

The radius of convergence r is a nonnegative real number or ∞ such that the series converges if

and diverges if

In other words, the series converges if z is close enough to the center and diverges if it is too far away. The radius of convergence specifies how close is close enough. On the boundary, that is, where |za| = r, the behavior of the power series may be complicated, and the series may converge for some values of z and diverge for others. The radius of convergence is infinite if the series converges for all complex numbers z.

Read more about this topic:  Radius Of Convergence

Famous quotes containing the word definition:

    I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.”
    Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.
    Ana Castillo (b. 1953)

    The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.
    William James (1842–1910)