1 Definition and basic form

A power series is an infinite sum built from powers of a variable centered at a fixed point. It is one of the standard ways to express a function locally, especially when the function is smooth or analytic. Because the terms are arranged by increasing degree, power series are well suited to algebraic manipulation and approximation.

1.1 General expression

The general power series has the form

\[ \sum_{n=0}^{\infty} a_n (x-c)^n, \]

where \(a_n\) is the coefficient of the \(n\)th term, \(x\) is the variable, and \(c\) is the center. The expression may converge for some values of \(x\) and diverge for others. When it converges, it defines a function on its domain of convergence.

1.2 Coefficients and center

The coefficients \(a_n\) determine the shape of the series and its behavior near the center \(c\). The center is the point around which the powers are formed, and changing it alters the series representation. In many settings, the coefficients are chosen to match derivatives of a function at that point.

1.3 Special cases

Certain familiar series are special instances of the general form. These examples often serve as prototypes for more complicated expansions and show how power series arise naturally in calculus and analysis.

1.3.1 Geometric series

The geometric series is the simplest power series, with terms that repeat a constant ratio. It is typically written as

\[ \sum_{n=0}^{\infty} x^n, \]

which converges for \(x<1\) and sums to \(1/(1-x)\). This example is important because many other series can be derived from it by algebraic or calculus-based operations.

1.3.2 Taylor series as power series

A Taylor series is a power series whose coefficients are determined by the derivatives of a function at a point. For a function \(f\) expanded about \(c\), the coefficients are

\[ a_n = \frac{f^{(n)}(c)}{n!}. \]

This construction links local behavior of a function to an infinite polynomial-like expansion.

1.3.3 Maclaurin series

A Maclaurin series is a Taylor series centered at \(0\). It has the form

\[ \sum_{n=0}^{\infty} a_n x^n. \]

Because the center is zero, it is especially convenient for many standard functions and for algebraic simplification.

2 Convergence

Convergence is the key issue for power series. A given series may converge for all \(x\), only near its center, or at no point except perhaps the center itself. The set of convergence is determined by a radius and an interval.

2.1 Radius of convergence

The radius of convergence is the nonnegative number \(R\) such that the series converges absolutely when \(x-c<R\) and diverges when \(x-c>R\). The case \(x-c=R\) must be checked separately. The radius may be zero or infinite.

2.1.1 Cauchy–Hadamard formula

The Cauchy–Hadamard formula gives the radius through the coefficients:

\[

\frac{1}{R} = \limsup_{n\to\infty} \sqrt[n]{a_n}.

\]

This formula is especially useful when direct comparison is difficult.

2.1.2 Ratio test

The ratio test examines the limit of \(a_{n+1}/a_n\), when it exists. For power series, it often provides a quick way to determine where the series converges absolutely. It is commonly applied to series with factorials or products in the coefficients.

2.1.3 Root test

The root test studies \(\sqrt[n]{a_n}\) and is closely related to the Cauchy–Hadamard formula. It is effective for coefficients with exponential or power growth. Like the ratio test, it identifies the radius of convergence.

2.2 Interval of convergence

For real power series, the interval of convergence is the set of real \(x\) values where the series converges. It usually has the form \((c-R, c+R)\), possibly with one or both endpoints included after separate testing. The interval summarizes the real domain on which the series defines a function.

2.3 Endpoint behavior

The boundary points often require individual analysis, since convergence at the ends is not guaranteed by the radius alone. Different endpoints may behave differently even for the same series. Careful testing is necessary there.

2.3.1 Absolute convergence

Absolute convergence occurs when the series of absolute values converges. This is the strongest common form of convergence for power series and guarantees stability under many operations. Inside the radius of convergence, power series converge absolutely.

2.3.2 Conditional convergence

Conditional convergence means a series converges but not absolutely. Power series can exhibit this behavior at endpoints of the interval of convergence. Such cases often require specific convergence tests rather than coefficient-based formulas.

2.3.3 Divergence at endpoints

At an endpoint, a power series may diverge even if it converges throughout the interior interval. This is typical when the terms fail to approach zero sufficiently fast. Endpoint divergence is one reason the boundary must be checked separately.

3 Algebraic operations

Power series can often be combined in ways similar to finite polynomials. Within their common domain of convergence, standard algebraic operations produce new power series. These operations make them especially useful in symbolic and analytic work.

3.1 Addition and subtraction

Two power series with the same center can be added or subtracted term by term. The new coefficients are obtained by combining the corresponding coefficients of each series. The resulting series usually converges on at least the intersection of the original convergence domains.

3.2 Multiplication

The product of two power series is formed by distributing terms and collecting equal powers of \(x-c\). This produces coefficients that depend on sums of products from the original series. Multiplication is fundamental in generating new expansions from known ones.

3.2.1 Cauchy product

The Cauchy product is the standard rule for multiplying series. If two series are multiplied, the coefficient of each power is the sum of all pairwise coefficient products whose indices add to that power. Under appropriate convergence conditions, this gives the correct product series.

3.2.2 Product of convergent power series

When two power series both converge in a region, their product usually converges in a common subregion. The product then represents the pointwise product of the summed functions. This property supports the algebra of analytic functions.

3.3 Division

A power series may sometimes be divided by another series, provided the denominator has a nonzero constant term. The coefficients of the quotient can be found recursively. This process is analogous to long division of polynomials, but extended to infinite series.

3.4 Composition

Composition replaces the variable in one power series with another power series, when the substitution stays within the domain of convergence. This operation is more delicate than addition or multiplication because convergence must be controlled carefully. When valid, it generates complicated expansions from simpler ones.

3.5 Reindexing and substitution

Reindexing rewrites a series using a different summation index, often to simplify formulas or align powers. Substitution changes the variable in a controlled way, such as replacing \(x\) by \(ax+b\) or by another series under suitable conditions. These techniques are common in derivations and coefficient matching.

4 Calculus of power series

Power series behave especially well under differentiation and integration. Within the interval of convergence, these operations may be performed term by term. This makes power series a powerful tool for turning analytic problems into algebraic ones.

4.1 Term-by-term differentiation

A power series may be differentiated by differentiating each term separately. The differentiated series has coefficients multiplied by the corresponding exponent and powers reduced by one. The result converges at least on the same open interval as the original series.

4.2 Term-by-term integration

A power series may also be integrated term by term. Integration raises each power by one and divides by the new exponent. This operation preserves convergence within the same open interval and is often used to derive new closed forms.

4.3 Higher-order derivatives

Repeated differentiation of a power series remains valid inside the interval of convergence. Each derivative can be obtained by differentiating the preceding series term by term. At the center, the derivatives recover the coefficients of the series in a direct way.

4.4 Preservation of radius of convergence

Differentiation and integration do not change the radius of convergence. Although endpoint behavior may vary after these operations, the interior convergence region remains the same. This stability is one of the most useful features of power series.

4.5 Analyticity on the interval of convergence

A function represented by a power series is analytic wherever the series converges. In such regions, the function equals its own local expansion and can be recovered from its coefficients. This property underlies much of classical analysis and complex function theory.

5 Representation of functions

Power series are important because they can represent functions by local data. When such a representation exists, the series encodes derivatives, approximations, and structural information about the function. Not every function admits one, but many standard functions do.

5.1 Taylor’s theorem

Taylor’s theorem relates a function to a polynomial approximation and a remainder term. If the remainder tends to zero under suitable conditions, the Taylor polynomial sequence converges to a Taylor series. This theorem provides the bridge between finite approximation and infinite expansion.

5.2 Conditions for a function to equal its power series

A function equals its power series when the Taylor series converges to the function at each point in a region. Smoothness alone is not always sufficient; the remainder must vanish appropriately. For analytic functions, this equality holds in a neighborhood of the center.

5.3 Uniqueness of coefficients

If a function has a power series expansion about a point, its coefficients are uniquely determined by the function. In particular, matching all derivatives at the center fixes the entire series. This uniqueness is a key reason power series are powerful in function theory.

5.4 Approximation by partial sums

Truncating a power series gives a partial sum, which is a polynomial approximation to the original function. Partial sums are often used in computation because they are easier to evaluate than infinite series. The approximation improves as more terms are included, within the convergence region.

6 Common examples

Many widely used functions have standard power series expansions. These examples appear repeatedly in calculus, algebra, and applied mathematics. They also provide templates for deriving new series.

6.1 Exponential function

The exponential function has the Maclaurin series

\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}. \]

This series converges for all real and complex \(x\). Its rapid convergence makes it especially useful in computation.

6.2 Trigonometric functions

The sine and cosine functions have expansions centered at zero:

\[ \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}, \quad \cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}. \]

These series converge for all \(x\) and are central in analysis and physics.

6.3 Logarithmic function

The natural logarithm can be expanded near \(x=0\) through

\[ \ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1}\frac{x^n}{n}, \]

which converges for \(-1 < x \le 1\) with endpoint behavior determined separately. This is a standard example of a series with a finite radius of convergence.

6.4 Binomial series

For general exponent \(\alpha\), the binomial series gives

\[ (1+x)^\alpha = \sum_{n=0}^{\infty} {\alpha \choose n} x^n, \]

where the generalized binomial coefficients are defined using falling products and factorials. It converges for \(x<1\) in general. This expansion extends polynomial binomial formulas to noninteger powers.

6.5 Rational functions

Many rational functions can be expanded as power series near suitable centers. For example, a function of the form \(1/(1-x)\) or \(1/(a-x)\) can often be written using the geometric series. Such expansions are useful for local analysis and coefficient extraction.

7 Applications

Power series are used widely because they turn difficult functions into manageable algebraic forms. They provide local models, solve equations, and support numerical methods. Their influence extends across both pure and applied mathematics.

7.1 Solving differential equations

Series methods can solve differential equations by assuming a power series form for the unknown function. Substituting the series into the equation produces recurrence relations for the coefficients. This technique is especially useful near ordinary points and regular singular points.

7.2 Generating functions

In combinatorics, power series serve as generating functions that encode sequences. Coefficients represent counting data, and algebra on series corresponds to operations on the underlying sequences. This viewpoint is a major bridge between discrete mathematics and analysis.

7.3 Asymptotic approximations

Truncated power series can approximate functions near a point and provide asymptotic information. Even when a full series does not converge globally, its initial terms may still give accurate local estimates. Such approximations are common in perturbation methods and applied modeling.

7.4 Numerical computation

Power series allow functions to be computed by summing finitely many terms. This approach is practical when direct evaluation is difficult or when hardware implementations require polynomial approximations. Error control depends on the remainder after truncation.

7.5 Complex analysis

In complex analysis, power series define analytic functions on discs of convergence. They are the local building blocks of holomorphic theory and connect naturally to contour methods and residue calculations. Many central results in complex analysis depend on series representations.

Power series are part of a broader family of series expansions. Several related concepts extend, generalize, or complement the idea of a polynomial-like infinite sum.

8.1 Formal power series

A formal power series is treated as an algebraic object without concern for convergence. Its coefficients are manipulated symbolically, which is useful in combinatorics and algebra. Unlike analytic power series, it is not initially interpreted as a function.

8.2 Laurent series

A Laurent series allows both positive and negative powers of the variable. It is especially useful near isolated singularities in complex analysis. Power series may be viewed as a special case with no negative powers.

8.3 Analytic continuation

Analytic continuation extends a function beyond the region where a given power series converges. By overlapping local expansions, one can often enlarge the domain of an analytic function. This process is central to complex function theory.

8.4 Fourier series

A Fourier series represents a periodic function as a sum of sines and cosines rather than powers. Although structurally different from a power series, it shares the goal of function approximation by infinite series. The two theories are often compared in analysis and applied mathematics.

&lt;/INTERNAL_LINK_CANDIDATES&gt; Geometric series (a basic power series with constant ratio) Taylor series (a power series determined by derivatives at a point) Maclaurin series (a Taylor series centered at zero) Radius of convergence (the size of the region where a power series converges) Cauchy–Hadamard formula (a formula for computing radius of convergence) Ratio test (a convergence test using successive coefficient ratios) Root test (a convergence test using nth roots of coefficients) Interval of convergence (the set of real inputs where the series converges) Absolute convergence (convergence after taking absolute values) Conditional convergence (convergence without absolute convergence) Cauchy product (the coefficient rule for multiplying series) Formal power series (an algebraic power series without convergence requirements) Laurent series (a series with positive and negative powers) Analyticity (the property of being representable by a convergent power series) Taylor’s theorem (the theorem relating functions to Taylor expansions) Generating function (a series encoding a sequence) Analytic continuation (extension of an analytic function beyond a given domain) Fourier series (an expansion in sines and cosines) Partial sums (finite truncations used to approximate a series) Differential equation (an equation involving derivatives, often solved by power series)