Power Series and Radius of Convergence

A power series is a function represented as an infinite sum of terms involving powers of $x$, and determining precisely where this infinite sum adds up to a finite number is the cornerstone of using it for approximation, analysis, and solving differential equations. Mastering the radius of convergence and the behavior at the interval's endpoints transforms a mysterious infinite series into a precise and powerful tool for representing functions.

What is a Power Series?

Formally, a power series centered at a point $a$ is an expression of the form: $$\sum _{n=0}^{\infty }{c}_n(x-a{)}^n=c_0+c_1(x-a)+c_2(x-a{)}^2+\cdots$$ Here, the coefficients $c_n$ are constants, $a$ is the center, and $x$ is the variable. When $a=0$, the series simplifies to ${\sum }_{n=0}^{\infty }{c}_n{x}^n$. The fundamental question is: for which values of $x$ does this infinite sum converge to a finite number? The answer is always an interval of convergence centered at $x=a$.

The radius of convergence, denoted by $R$, is a non-negative number or infinity that defines the distance from the center $a$ within which the series converges absolutely. The set of all $x$ for which the series converges—the interval of convergence—is then $(a-R,a+R)$, possibly including one or both endpoints. For $|x-a|>R$, the series diverges.

Finding the Radius and Interval of Convergence

Two primary tests are used to find $R$: the Ratio Test and the Root Test. They are applied to the absolute value of the series' terms.

The Ratio Test is the most common tool. For a power series $\sum c_n(x-a{)}^n$, you compute the limit $$L=\underset{n\to \infty }{\lim }\left|\frac{c_{n+1}(x-a{)}^{n+1}}{c_n(x-a{)}^n}\right|=|x-a|\underset{n\to \infty }{\lim }\left|\frac{c_{n+1}}{c_n}\right|.$$ By the Ratio Test, the series converges absolutely when $L<1$. This inequality, $|x-a|\cdot \lim |c_{n+1}/c_n|<1$, directly solves to $|x-a|<R$, where the radius is $$R=\frac{1}{{\lim }_{n\to \infty }\left|\frac{c_{n+1}}{c_n}\right|},$$ provided this limit exists. If the limit is 0, then $R=\infty$; if the limit is infinity, then $R=0$.

The Root Test is similarly applied. Compute $$L=\underset{n\to \infty }{\lim }\sqrt[n]{|c_n(x-a{)}^n|}=|x-a|\underset{n\to \infty }{\lim }\sqrt[n]{|c_n|}.$$ Again, convergence is guaranteed when $L<1$, yielding $|x-a|<R$ with $$R=\frac{1}{{\lim }_{n\to \infty }\sqrt[n]{|c_n|}}.$$

Example: Find the radius and interval of convergence for ${\sum }_{n=1}^{\infty }\frac{x^n}{n}$. We use the Ratio Test: $$\underset{n\to \infty }{\lim }\left|\frac{x^{n+1}}{n+1}\cdot \frac{n}{x^n}\right|=|x|\underset{n\to \infty }{\lim }\frac{n}{n+1}=|x|.$$ The series converges absolutely when $|x|<1$, so $R=1$. The interval of convergence is centered at 0 with radius 1: $(-1,1)$. We must check the endpoints separately. At $x=1$, the series is the harmonic series $\sum 1/n$, which diverges. At $x=-1$, the series is the alternating harmonic series $\sum (-1{)}^n/n$, which converges conditionally. Therefore, the interval of convergence is $[-1,1)$.

Term-by-Term Operations

One of the most powerful features of power series within their interval of convergence is that they behave like polynomials. Specifically, within the open interval $|x-a|<R$:

1. Term-by-Term Differentiation: The derivative of the series is the series of the derivatives.

$$f(x)=\sum _{n=0}^{\infty }{c}_n(x-a{)}^n\Rightarrow f^{\prime }(x)=\sum _{n=1}^{\infty }n{c}_n(x-a{)}^{n-1}.$$ The new series also has radius of convergence $R$.

1. Term-by-Term Integration: The integral of the series is the series of the integrals.

$$\int f(x)dx=C+\sum _{n=0}^{\infty }\frac{c_n}{n+1}(x-a{)}^{n+1}.$$ The new series also has radius of convergence $R$.

These properties are foundational for solving differential equations with series methods and for finding new power series representations from known ones. For example, starting with the geometric series $\frac{1}{1-x}={\sum }_{n=0}^{\infty }{x}^n$ for $|x|<1$, you can integrate term-by-term to find the series for $-\ln |1-x|$.

Boundary Behavior and Abel's Theorem

The behavior of a power series at the endpoints $x=a\pm R$ of its interval of convergence can be tricky, involving conditional convergence or divergence. Abel's Theorem provides a crucial insight about continuity at a convergent endpoint.

Abel's Theorem states: If the power series $\sum c_n(x-a{)}^n$ converges at the right endpoint $x=a+R$ (where $R$ is the radius of convergence), then the series converges uniformly on the interval $[a,a+R]$. Consequently, the function $f(x)$ defined by the series is left-continuous at this endpoint: $$\underset{x\to (a+R{)}^{-}}{\lim }f(x)=\sum _{n=0}^{\infty }{c}_n{R}^n.$$ An analogous result holds for the left endpoint.

This theorem is vital for evaluating sums of convergent numerical series. For instance, we know the series for $\ln (1+x)$ is $x-x^2/2+x^3/3-\cdots$ with $R=1$ and it converges at $x=1$ (alternating harmonic series). Abel's Theorem guarantees that as $x\to 1^{-}$, the series sum approaches $\ln (2)$, confirming that $1-1/2+1/3-\cdots =\ln (2)$.

Applications: Representing Functions as Power Series

The ultimate utility of power series lies in representing complex functions with infinite polynomials, enabling approximation, integration, and computation.

• Analytic Functions: A function $f$ is analytic at a point $a$ if it can be represented by a power series converging on some interval around $a$. Common examples include $e^x$, $\sin x$, and $\cos x$, which have series convergent for all $x$ (i.e., $R=\infty$).
• Taylor and Maclaurin Series: The specific power series representation of a function $f$ about $a$ is given by its Taylor series:

$$\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n.$$ Finding the radius of convergence of this series tells you the domain on which this representation of $f$ is valid.

• Solving Differential Equations: Many differential equations without elementary solutions can be solved by assuming a solution in the form of a power series $y=\sum c_n{x}^n$, substituting into the equation, and solving for the coefficients term-by-term. The first step is always to find the radius of convergence for the resulting series solution.

Common Pitfalls

1. Confusing Radius and Interval of Convergence: The radius $R$ only gives the open interval $(a-R,a+R)$. You must test each endpoint $x=a\pm R$ separately using other convergence tests (like the Alternating Series Test or p-series test) to determine the full interval of convergence. A series may converge at neither, one, or both endpoints.
2. Misapplying the Ratio/Root Test Formula: The formulas $R=1/\lim |c_{n+1}/c_n|$ and $R=1/\lim \sqrt[n]{|c_n|}$ are derived from the condition $L<1$. If this limit is zero, $R$ is infinite. If the limit does not exist, you must use the full limit definition ($\mathrm{limsup}$ for the Root Test) rather than a simple formula.
3. Assuming Operations Preserve Convergence at Endpoints: Term-by-term differentiation and integration preserve the radius of convergence $R$, but they can alter behavior at the endpoints. A series might converge at an endpoint, but its derivative series might diverge there. Always re-check endpoints after differentiation or integration.
4. Overlooking Conditional Convergence: At an endpoint, a series may converge conditionally (like the alternating harmonic series). When dealing with such series, the rules for finite sums do not always apply; rearranging terms can change the sum. Abel's Theorem is valuable here because it links the sum of the series at a conditionally convergent endpoint to the limit of the function from inside the interval.

Summary

• A power series $\sum c_n(x-a{)}^n$ converges within a symmetric interval of convergence centered at $a$, defined by its radius of convergence $R$.
• The Ratio Test and Root Test are the primary tools for finding $R$. The interval is found by testing the convergence of the series at each endpoint $x=a\pm R$ separately.
• Within the open interval $|x-a|<R$, power series can be differentiated and integrated term-by-term to yield new series with the same radius of convergence.
• Abel's Theorem describes the continuous behavior of a power series function at a convergent endpoint, providing a rigorous link between the series sum and the limit of the function.
• Power series representations (like Taylor series) are essential for working with analytic functions, performing approximations, integrating non-elementary functions, and finding series solutions to differential equations.