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: $n = 0 \sum \infty c_{n} (x - a)^{n} = c_{0} + c_{1} (x - a) + c_{2} (x - a)^{2} + \dots$ 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 $L = n \to \infty lim \frac{c _{n + 1} ( x - a ) ^{n + 1}}{c _{n} ( x - a ) ^{n}} = ∣ x - a ∣ n \to \infty lim \frac{c _{n + 1}}{c _{n}} .$ 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 $R = \frac{1}{lim _{n \to \infty} \frac{c _{n + 1}}{c _{n}}},$ 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 $L = n \to \infty lim n ∣ c_{n} (x - a)^{n} ∣ = ∣ x - a ∣ n \to \infty lim n ∣ c_{n} ∣ .$ Again, convergence is guaranteed when $L < 1$ , yielding $∣ x - a ∣ < R$ with $R = \frac{1}{lim _{n \to \infty} n ∣ c _{n} ∣} .$

Example: Find the radius and interval of convergence for $\sum_{n = 1}^{\infty} \frac{x ^{n}}{n}$ . We use the Ratio Test: $n \to \infty lim \frac{x ^{n + 1}}{n + 1} \cdot \frac{n}{x ^{n}} = ∣ x ∣ n \to \infty lim \frac{n}{n + 1} = ∣ x ∣.$ 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 $∣ x - a ∣ < R$ :

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

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

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

$\int f (x) d x = C + n = 0 \sum \infty \frac{c _{n}}{n + 1} (x - a)^{n + 1} .$ 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: $x \to (a + R)^{-} lim f (x) = n = 0 \sum \infty c_{n} R^{n} .$ 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 $x - x^{2} /2 + x^{3} /3 - \dots$ 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 $1 - 1/2 + 1/3 - \dots = ln (2)$ .

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:

$n = 0 \sum \infty \frac{f ^{(n)} ( a )}{n !} (x - a)^{n} .$ 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

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.
Misapplying the Ratio/Root Test Formula: The formulas $R = 1/ lim ∣ c_{n + 1} / c_{n} ∣$ and $R = 1/ lim n ∣ c_{n} ∣$ 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 ( $lim sup$ for the Root Test) rather than a simple formula.
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.
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.

Power Series and Radius of Convergence

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 $$L = \lim_{n \to \infty} \left| \frac{c_{n+1}(x-a)^{n+1}}{c_n (x-a)^n} \right| = |x-a| \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right|.$$ 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 $$L = \lim_{n \to \infty} \sqrt[n]{|c_n (x-a)^n|} = |x-a| \lim_{n \to \infty} \sqrt[n]{|c_n|}.$$ 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: $$\lim_{n \to \infty} \left| \frac{x^{n+1}}{n+1} \cdot \frac{n}{x^n} \right| = |x| \lim_{n \to \infty} \frac{n}{n+1} = |x|.$$ 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 $|x-a| < R$ :

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

$$f(x) = \sum_{n=0}^{\infty} c_n (x-a)^n \quad \Rightarrow \quad f'(x) = \sum_{n=1}^{\infty} n c_n (x-a)^{n-1}.$$ The new series also has radius of convergence $R$ .

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

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: $$\lim_{x \to (a+R)^{-}} f(x) = \sum_{n=0}^{\infty} c_n R^n.$$ 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

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.
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 ( $\limsup$ for the Root Test) rather than a simple formula.
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.
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.

Power Series and Radius of Convergence

Power Series and Radius of Convergence

What is a Power Series?

Finding the Radius and Interval of Convergence

Term-by-Term Operations

Boundary Behavior and Abel's Theorem

Applications: Representing Functions as Power Series

Common Pitfalls

Summary

Power Series and Radius of Convergence

What is a Power Series?

Finding the Radius and Interval of Convergence

Term-by-Term Operations

Boundary Behavior and Abel's Theorem

Applications: Representing Functions as Power Series

Common Pitfalls

Summary

Write better notes with AI

Power Series and Radius of Convergence

What is a Power Series?

Finding the Radius and Interval of Convergence

Term-by-Term Operations

Boundary Behavior and Abel's Theorem

Applications: Representing Functions as Power Series

Common Pitfalls

Summary