1 Definition and basic ideas
The radius of convergence is a numerical measure attached to a power series. It describes how far one may move from the expansion point while still retaining convergence of the infinite sum. For a series centered at a point, the radius determines a region in which the series defines a function in a reliable and stable way.
1.1 Power series
A power series has the form \[ \sum_{n=0}^{\infty} a_n (x-c)^n, \] where \(c\) is the center, \(a_n\) are coefficients, and \(x\) is the variable. Such series appear throughout calculus and analysis because they provide a compact way to represent functions by infinitely many polynomial terms.
Power series behave differently from ordinary polynomials because their convergence depends on the value of \(x\). Near the center, the terms may become small enough to sum to a finite limit, while farther away the same series may fail to converge.
1.2 Convergence behavior
For a given power series, convergence is not usually all-or-nothing across the real line or complex plane. Instead, there is typically a threshold distance from the center. Inside that range, the series converges; outside it, the series diverges.
1.2.1 Absolute convergence
Within the radius of convergence, a power series converges absolutely. This means the series formed by the absolute values of its terms also converges. Absolute convergence is especially useful because it implies strong stability properties and allows many termwise operations.
1.2.2 Divergence outside the radius
Beyond the radius of convergence, the power series diverges. At such points, the terms do not approach zero quickly enough for the sum to settle to a finite value. The boundary between convergence and divergence is usually sharp, though the behavior exactly on the boundary may vary.
1.3 Interval and disk of convergence
For real power series, the set of convergence is typically an interval on the real line centered at \(c\). This interval may be open, closed, or half-open at the endpoints depending on boundary behavior.
For complex power series, the corresponding region is a disk in the complex plane centered at the expansion point. The radius of that disk is the radius of convergence, and points outside the disk do not belong to the convergence region.
2 Formal definition
The radius of convergence is defined as the largest nonnegative number \(R\) such that the power series converges for all points with distance less than \(R\) from the center, and diverges for all points with distance greater than \(R\). If the series converges only at the center, the radius is zero. If it converges everywhere, the radius is infinite.
2.1 Radius in real analysis
| In real analysis, a power series centered at \(c\) converges for all real \(x\) satisfying \( | x-c | <R\). The interval of convergence may extend to one or both endpoints, but the radius itself depends only on the interior region where convergence is guaranteed. |
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This notion is fundamental in studying functions locally around a point. It tells how far the series representation can be trusted as a direct description of the function.
2.2 Radius in complex analysis
In complex analysis, the variable is a complex number \(z\), and the power series takes the form \[ \sum_{n=0}^{\infty} a_n (z-c)^n. \] The radius of convergence describes a geometric region in the complex plane rather than an interval on the real line.
2.2.1 Convergence disks in the complex plane
| If \(R>0\), the series converges absolutely for all \(z\) with \( | z-c | <R\). This set is an open disk centered at \(c\). The function defined by the series is analytic throughout that disk, and the series may be differentiated or integrated term by term inside it. |
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2.2.2 Boundary behavior
| On the boundary \( | z-c | =R\), convergence must be checked separately. Some power series converge at every boundary point, some at none, and many exhibit mixed behavior. The radius gives no automatic conclusion there, only a clear division between the interior and exterior regions. |
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3 Methods of determining the radius
Several standard tests can be used to compute or estimate the radius of convergence. These methods are based on the size and growth rate of the coefficients.
3.1 Ratio test
If the limit \[
| \lim_{n\to\infty}\left | \frac{a_{n}}{a_{n+1}}\right |
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\] exists, it often gives the radius directly. More commonly, one applies the ratio test to the terms \(a_n(x-c)^n\). When the ratio of successive terms approaches a limit, that limit determines the threshold for convergence.
The ratio test is especially effective when coefficients have a simple recursive pattern or factorial structure.
3.2 Root test
| The root test examines the quantity \(\sqrt[n]{ | a_n(x-c)^n | }\). If the limit of \(\sqrt[n]{ | a_n | }\) exists, it can be used to determine the radius. This test is often convenient when coefficients involve exponentials or powers that behave regularly under \(n\)th roots. |
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3.3 Cauchy–Hadamard formula
The Cauchy–Hadamard formula gives the radius in terms of the upper limit behavior of the coefficients. It is one of the most general tools for power series.
3.3.1 limsup formulation
The formula states that \[
| \frac{1}{R}=\limsup_{n\to\infty}\sqrt[n]{ | a_n | }, |
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\] with the conventions that \(R=\infty\) if the limsup is \(0\), and \(R=0\) if the limsup is infinite. This expression captures the asymptotic growth of the coefficients even when ordinary limits do not exist.
3.4 Comparison with known series
Sometimes a radius can be found by comparing the given series to a standard one whose convergence properties are already known. If the coefficients resemble those of a geometric, exponential, or binomial series, the comparison may quickly reveal the same type of convergence threshold.
4 Examples
Examples show how the abstract definition works in familiar cases. They also illustrate that different series can have very different radii, ranging from zero to infinity.
4.1 Geometric series
The geometric series \[ \sum_{n=0}^{\infty} z^n \]
| has radius of convergence \(R=1\). It converges when \( | z | <1\) and diverges when \( | z | \ge 1\). This series is a basic model for all power series because its convergence region is easy to describe. |
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4.2 Exponential function
The exponential series \[ \sum_{n=0}^{\infty}\frac{z^n}{n!} \] has radius of convergence \(R=\infty\). The factorial in the denominator grows so rapidly that the terms become small for every complex number \(z\). As a result, the series converges everywhere.
4.3 Logarithmic series
The series \[ \sum_{n=1}^{\infty}\frac{(-1)^{n+1}z^n}{n} \]
| has radius of convergence \(R=1\). It is related to the expansion of \(\log(1+z)\) near \(z=0\). The series converges for \( | z | <1\), while its behavior at the boundary must be examined point by point. |
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4.4 Binomial series
For a general exponent \(\alpha\), the binomial series \[ \sum_{n=0}^{\infty} \binom{\alpha}{n} z^n \] also has radius of convergence \(R=1\). It represents \((1+z)^\alpha\) near \(z=0\) when suitable branches are chosen. This example is important because it connects algebraic expressions with analytic expansions.
5 Properties
The radius of convergence has several useful structural properties. It is determined uniquely by the coefficients, and it behaves predictably under many common operations on series.
5.1 Uniqueness of the radius
A power series has exactly one radius of convergence. No other number can play the same role, because the set of points where the series converges is fixed once the coefficients are fixed. This uniqueness makes the radius a well-defined invariant of the series.
5.2 Behavior under algebraic operations
Operations on power series often produce new series whose radii can be related to those of the original series. In many cases, the new radius is at least as large as expected from the factors involved.
5.2.1 Addition and subtraction
When two power series are added or subtracted, the resulting series has a radius of convergence at least as large as the smaller of the two radii. The region where both original series converge is automatically a region of convergence for the sum or difference.
5.2.2 Multiplication and division
The product of two power series usually converges in the overlap of their convergence regions, so its radius is at least the minimum of the two. Division is more delicate and requires that the denominator series not vanish at the center; under suitable conditions, a quotient series may be formed with a corresponding convergence radius.
5.3 Termwise differentiation and integration
Within the disk or interval of convergence, a power series may be differentiated or integrated term by term. These operations preserve the radius of convergence. In particular, the differentiated and integrated series converge on the same interior region as the original one, though endpoint behavior can differ in the real case.
6 Relation to Taylor and analytic functions
Radius of convergence is closely tied to Taylor series and the theory of analytic functions. It provides a precise measure of how locally a function can be represented by its derivatives at a point.
6.1 Taylor series expansions
A Taylor series is a power series built from the derivatives of a function at a center point. The radius of convergence tells where that Taylor expansion actually converges to a function. In many classical cases, the Taylor series agrees with the original function throughout its convergence disk or interval.
6.2 Analyticity and local representability
A function is analytic at a point if it can be represented by a convergent power series in some neighborhood of that point. The radius of convergence is therefore a measure of the size of the neighborhood on which the function has such a representation. A larger radius means a broader local analytic description.
6.3 Singularities and distance to the nearest singularity
For many analytic functions, the radius of convergence equals the distance from the center to the nearest singularity in the complex plane. This principle explains why the singularity structure of a function strongly influences its series expansion. If the nearest obstruction is far away, the series converges in a larger region.
7 Extensions and advanced topics
The idea of a radius of convergence extends beyond elementary power series. Related notions appear in generalized series, several complex variables, and analytic continuation.
7.1 Radius for Laurent series
A Laurent series includes negative as well as nonnegative powers. Instead of a disk, its convergence region is often an annulus. The two boundary radii describe an inner and outer limit, beyond which the series fails to converge.
7.2 Multivariable power series
In several variables, power series may converge in regions determined by more than one scale. The simple notion of a single radius is replaced by a convergence domain, often described using norms or polydisks. These regions reflect the geometry of the variables and their combined growth.
7.3 Growth estimates and analytic continuation
The radius of convergence is also connected to coefficient growth and to analytic continuation. Knowing the coefficient behavior can reveal how far the function extends as an analytic object. When a function can be continued beyond its initial disk, the original radius marks only the first local domain, not necessarily the full extent of the function.