In mathematics, the radius of convergence of a power series Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

where the center a and the coefficients c_n are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i represents the imaginary number, i2 = −1. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

and diverges if

In other words, the series converges if z is close enough to the center. The radius of convergence specifies how close is close enough. The radius of convergence is infinite if the series converges for all complex numbers z. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i represents the imaginary number, i2 = −1. ...

Existence and value of the radius of convergence

The radius of convergence can be found by applying the root test to the terms of the series. If In mathematics, the root test is a test for the convergence of an infinite series. ...

(where "lim sup" denotes the limit superior), then the radius of convergence is 1/C. If C = 0, then the radius of convergence is infinite, meaning that f is an entire function. In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. ... In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. ...

In many cases, the ratio test suffices. If the limit In mathematics, the ratio test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. ...

exists, then the radius of convergence is 1/L. This limit is often easier to compute than the limit for C above. However, it may not exist, in which case one has to use the formula for C instead.

Clarity and simplicity result from complexity

One of the best examples of clarity and simplicity following from thinking about complex numbers where confusion would result from thinking about real numbers is this theorem of complex analysis: Complex analysis is the branch of mathematics investigating functions of complex numbers. ...

The radius of convergence is always equal to the distance from the center to the nearest point where the function f has a (non-removable) singularity; if no such point exists then the radius of convergence is infinite.

The nearest point means the nearest point in the complex plane, not necessarily on the real line, even if the center and all coefficients are real. See holomorphic functions are analytic; the result stated above is a by-product of the proof found in that article. In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... In complex analysis, a complex-valued function f of a complex variable is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a, and is analytic at a if in some open disk centered at a it can be expanded as...

A simple warm-up example

The arctangent function of trigonometry can be expanded in a power series familiar to calculus students: Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. ...

It is easy to apply the ratio test in this case to find that the radius of convergence is 1. But we can also view the matter thus:

and a zero appears in the denominator when z² = − 1, i.e., when z = i or − i. The center in this power series is at 0. The distance from 0 to either of these two singularities is 1. That is therefore the radius of convergence.

(This famous example also immediately gives us a method for calculating the value of π. It is an interesting application of Abel's theorem. In view of Leibniz' test (described in the entry alternating series) the series In real analysis, Abels theorem for power series relates a limit of a power series to the sum of its coefficients. ... In mathematics, an alternating series is an infinite series of the form with an ≥ 0. ...

converges. So Abel's theorem tells us that the sum of this series must equal

.

In view of Leibniz' theorem we can also easily determine how many terms of this series we need to use to find π to within any required accuracy. For a slightly different explanation of this calculation see the entry Leibniz formula for pi.) In mathematics, Leibniz formula for Ï€, due to Gottfried Leibniz, states that Proof Consider the infinite geometric series It is the limit of the truncated geometric series Splitting the integrand as and integrating both sides from 0 to 1, we have Integrating the first integral (over the truncated geometric series ) termwise...

A gaudier example

Consider this power series:

where the coefficients B_n are the Bernoulli numbers. It may be cumbersome to try to apply the ratio test to find the radius of convergence of this series. But the theorem of complex analysis stated above quickly solves the problem. At z = 0, there is in effect no singularity since the singularity is removable. The only non-removable singularities are therefore located where the denominator is zero. We solve In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... In complex analysis, a removable singularity of a function is a point at which the function is not defined (a singularity) but at which the function can be defined without creating any problems. ...

e^z − 1 = 0

by recalling that if z = x + iy and e^iy = cos(y) + i sin(y) then

e^z = e^xe^iy = e^x(cos(y) + isin(y)),

and then take x and y to be real. Since y is real, the absolute value of cos(y) + i sin(y) is necessarily 1. Therefore, the absolute value of e^z can be 1 only if e^x is 1; since x is real, that happens only if x = 0. Therefore we need cos(y) + i sin(y) = 1. Since y is real, that happens only if cos(y) = 1 and sin(y) = 0, so that y is an integral multiple of 2π. Since the real part x is 0 and the imaginary part y is a nonzero integral multiple of 2π, the solution of our equation is

z = a nonzero integral multiple of 2πi.

The singularity nearest the center (the center is 0 in this case) is at 2πi or − 2πi. The distance from the center to either of those points is 2π. That is therefore the radius of convergence.