Contents

The concept of how to find the **radius of convergence** is the heart of **power series** in **calculus**, which one cannot overlook. Acting as the boundary between **convergence** and **divergence**, the **radius of convergence** breathes life into power series by defining the set of **x-values** for which the **series converges**.

Whether you’re a student grappling with the foundations of **calculus** or an expert seeking to brush up on your knowledge, understanding how to find the **radius of convergence** is critical.

In the following article, we will demystify the process of finding this elusive yet essential mathematical parameter. From its **theoretical** underpinnings to the **nitty-gritty** of calculations, we’ll explore a variety of approaches to **efficiently** and **accurately** find the **radius of convergence** for a given power series.

**How to Find the Radius of Convergence**

To **find the radius of convergence**, apply the Ratio Test: Determine the limit of the absolute value of the ratio of consecutive coefficients as the term number approaches infinity. If the limit exists, that value is the radius of convergence.

The **radius of convergence** of a **power series** ∑aₙ(x – c)ⁿ (from n = 0 to infinity) is the value **r** such that the series converges for all **x** for which |**x – c**| < **r**, and diverges for all **x** for which |**x – c**| > **r**.

In simple terms, it’s the distance from the center ‘**c**‘ of the **power series** to the endpoints of the **interval** of **convergence**. Below in figure-1, we present a generic power series and its radius of convergence.

Figure-1.

## Techniques of How to Find the Radius of Convergence

**Ratio Test Method**

This is the most commonly used method to find the **radius of convergence**.

For the given **power series**, take the ratio of the (**n+1**)th term to the **nth** term in absolute values, take the limit as **n** approaches infinity, and set this limit to be less than 1. This gives you the interval of convergence.

The **ratio test** states that for a series **∑aₙ**, if we have **L = lim (n→∞) |aₙ₊₁/aₙ|**, the series converges absolutely if **L < 1**.

For the power series, this will yield an inequality of the form |**x – c**| < **r**, where **r** is the **radius of convergence**.

**Root Test Method**

Another method to find the **radius of convergence** uses the **root test**, which is particularly useful when the terms of the series have **nth roots** or powers of **n**.

For the given **power series**, take the **nth root** of the absolute value of the **nth** term, take the limit as **n** approaches infinity, and set this limit to be less than 1.

The **root test** states that for a series **∑aₙ**, if we have **L = lim (n→∞) |aₙ|⁽¹/ⁿ⁾**, the series converges absolutely if **L < 1**.

For the power series, this will also yield an inequality of the form |**x – c**| < **r**, where **r** is the **radius of convergence**.

Remember, these methods only give the **radius of convergence**. To fully determine the **interval of convergence**, you must also check whether the **series converges** at the **endpoints** **x = c ± r** by substituting these values into the series and applying one of the **convergence tests**.

**Historical Significance**

The concept of the **radius of convergence** is part of a larger mathematical field called **complex analysis**, which is an extension of **calculus**. The origins of this concept are tied to the development of complex analysis and the use of **power series** in the 18th and 19th centuries.

The use of **power series** dates back to the time of **Newton** and **Leibniz** in the late 17th century, with Newton using power series as a primary tool in his development of calculus. In these early days, however, the concept of a “**radius of convergence**” had not yet been established.

Instead, mathematicians were mainly concerned with whether a given power series **converged** or **diverged** for specific variable values.

It wasn’t until the 18th century that mathematicians established a complete theory of power series. Swiss mathematician **Leonhard Euler** was particularly influential, extensively using power series in his work. Although Euler did not explicitly define the radius of convergence, he implicitly used the concept in his manipulations of power series.

The term “**radius of convergence**” and the rigorous theory surrounding it came about in the 19th century as mathematicians began to formulate the field of complex analysis. French mathematician **Augustin-Louis Cauchy**, one of the key figures in the development of complex analysis, provided much of the groundwork.

Cauchy was the first to prove that a power series converges absolutely within its circle (or “disk”) of convergence, which directly relates to the concept of the **radius of convergence**.

**Karl Weierstrass**, a German mathematician, later provided a more general and rigorous formulation of the limit processes involved, including the formulation of the **root test**, which can be used to find the radius of convergence of a power series.

Today, the concept of the **radius of convergence** is a standard part of any course in complex analysis or advanced calculus, and it plays a crucial role in many areas of mathematics, physics, and engineering.

**Properties**

The **radius of convergence** is closely tied to the properties of **power series**, a fundamental type of series in calculus and analysis. Here are some key properties that pertain to finding the radius of convergence:

**Uniqueness**

For a given **power series**, there is exactly one **radius of convergence**. The series will converge for all **x** within this radius about the center **c** and will **diverge** for all **x** outside it.

**Dependence on Terms of Series**

The **radius of convergence** is determined by the coefficients of the series, i.e., the terms **aₙ**. It does not depend on the center **c** of the **series**.

**Determining Convergence**

The **radius of convergence** determines an interval around the center of the series (**c – r**, **c + r**) where the **series converges**. However, it does not give information about the **c – r** and **c + r** endpoints. The series may** converge** or **diverge**, or one endpoint may behave differently than the other at these points. Each **endpoint** needs to be checked separately.

**Role in Analytic Functions**

The **radius of convergence** of a power series defines the domain over which the function represented by the series is **analytic**. Within this interval, the function has a **power series** representation that **converges** to the function.

**Relation to Ratio or Root Test**

The **radius of convergence** can be found using the ratio test or the **root test**. In general, if **L = lim (n→∞) |aₙ₊₁/aₙ|** or **L = lim (n→∞) |aₙ|⁽¹/ⁿ⁾**, the radius of **convergence** **r** is given by **1/L**. If **L = 0**, the **radius of convergence** is** ∞** (the series converges for all x); if **L = ∞**, the **radius of convergence** is **0** (the series only converges at the center point x = c).

**Handling of Zero Radius**

If the **radius of convergence is zero**, the series only **converges** at the center** x = c**.

**Handling of Infinite Radius**

If the **radius of convergence** is infinite, the series **converges** for all **real numbers**.

**Algebraic Operations**

If two **power series** both have a positive **radius of convergence**, you can add them together, subtract one from the other, multiply them, or divide one by the other to form a new **power series**. The new series will also have a positive **radius of convergence**, although determining the exact value requires additional work.

**Applications **

The concept of the **radius of convergence** is integral to many areas of mathematics and its applications in diverse fields such as **physics**, **engineering**, **computer science**, and **economics**. Some notable applications include:

**Complex Analysis**

In **complex analysis**, the **radius of convergence** is fundamental in defining and working with **power series** representations of complex functions. For instance, when defining a function as a power series in complex variables, the **radius of convergence** helps specify the region of the complex plane in which the power series is valid.

**Differential Equations**

The **radius of convergence** is crucial when using **power series solutions** for **differential equations**. The interval determined by the **radius of convergence** is the domain on which the solution is valid.

**Physics**

In **physics**, the **radius of convergence** is used in **quantum mechanics** and **electrodynamics** when calculating approximations for various quantities using **perturbation theory**. It’s also used in **statistical mechanics** when dealing with **partition functions** and **thermodynamic potentials**.

**Engineering**

In **signal processing** and **control systems engineering**, the **radius of convergence** is used when applying the **Z-transform** in discrete-time systems and the **Laplace transform** in continuous-time systems.

**Computer Science**

In **algorithms** and **numerical analysis**, the **radius of convergence** can influence the choice of methods for numerical approximation, as it can indicate how well a power series will approximate a function over a particular interval.

**Economics**

In **economics**, the concept of **convergence** is often used in the context of infinite series to model various economic phenomena, and understanding the **radius of convergence** is critical to ensure the validity of these models.

**Probability Theory**

In **probability theory**, **generating functions** are often used to solve complex problems. These are power series, and understanding their **radius of convergence** is crucial to determining the domain over which these functions are useful.

**Exercise **

**Example 1**

Consider the power series **∑nⁿ * xⁿ** for n from **0** to **infinity**. Determine for which values of **‘x’** this series will **converge**. In other words, find the **radius of convergence** of this power series.

### Solution

Apply the Ratio Test:

L = lim (n→∞) |(n+1)⁽ⁿ⁺¹⁾ x⁽ⁿ⁺¹⁾ / nⁿ xⁿ|

L = lim (n→∞) |(n+1) x|

L = |x| lim (n→∞) (n+1)

L = ∞ for all x ≠ 0

So, the series only** converges** for **x = 0**, and the **radius of convergence r = 0**.

Figure-2.

**Example 2**

Consider the power series **∑xⁿ/n!** for **n** from **0** to **infinity** frequently appears in mathematical analyses. We want to know for which real numbers **‘x’** this series converges. Can you determine the **radius of convergence** of this series?

Apply the Ratio Test:

L = lim (n→∞) |x⁽ⁿ⁺¹⁾/(n+1)! xⁿ/n!|

L = lim (n→∞) |x/(n+1)|

L = 0 for all x.

So, the series **converges** for all **x**, and the** radius of convergence r = ∞**.

Figure-3.

### Solution

**Example 3**

We have a power series** ∑(n!*xⁿ)** for **n** from **0** to **infinity**. This series has a specific range of **‘x’** values for which it converges. Task is to find the **radius of convergence**, i.e., the range of **‘x’** values where this series converges.

### Solution

Apply the Ratio Test:

L = lim (n→∞) |(n+1)! x⁽ⁿ⁺¹⁾ / n! xⁿ|

L = lim (n→∞) |(n+1) x|

L = ∞ for all x ≠ 0

So, the series only** converges** for **x = 0**, and the **radius of convergence r = 0**.

**Example 4**

Given a power series **∑(xⁿ) / n²** for **n** from **1** to **infinity**, we want to discover the** ‘x’** values for which this **series converges**. Determine the **radius of convergence** for this series.

### Solution

Apply the Ratio Test:

L = lim (n→∞) |x⁽ⁿ⁺¹⁾/(n+1)² xⁿ/n²| =

L |x| lim (n→∞) (n^2/(n+1)^2)

L = |x|

The series **converges** for** |x| < 1**, so the **radius of convergence r = 1**.

Figure-4.

**Example 5**

Look at the power series **∑((2ⁿ) * xⁿ) / n** for **n** from **1** to **infinity**. We want to identify the values of **‘x’** for which this **series converges**. Calculate the **radius of convergence** of this series?

### Solution

Apply the Ratio Test:

L = lim (n→∞) |((2⁽ⁿ⁺¹⁾x⁽ⁿ⁺¹⁾)/(n+1)) * (n/(2ⁿ xⁿ))|

L = 2|x| lim (n→∞) (n/(n+1))

L = 2|x|

The series **converges** for **|x| < 1/2**, so the **radius of convergence** **r = 1/2**.

**Example 6**

Examine the power series **∑xⁿ / 2ⁿ** for n from 0 to infinity. We aim to find the **‘x’** values for which this series converges. Figure out the **radius of convergence** for this series?

### Solution

Apply the Ratio Test:

L = lim (n→∞) |x⁽ⁿ⁺¹⁾/(2⁽ⁿ⁺¹⁾) xⁿ/2ⁿ|

L = |x/2|

The series **converges** for **|x/2| < 1**, so the **radius of convergence r = 2**.

**Example 7**

Consider the power series **∑(n²) * xⁿ** for** n** from **0** to **infinity**. We are interested in the values of** ‘x’** for which this series converges. Find the **radius of convergence** of this power series.

### Solution

Apply the Ratio Test:

L = lim (n→∞) |((n+1)² x⁽ⁿ⁺¹⁾) / n² xⁿ|

L = |x| lim (n→∞) ((n+1)² / n²)

L = |x|

The series **converges** for **|x| < 1**, so the **radius of convergence** **r = 1**.

**Example 8**

Given the power series **∑(((-1)ⁿ) * xⁿ) / √n** for **n** from **1** to **infinity**, we want to find out the** ‘x’** values for which this series converges. Determine the** radius of convergence** of this series?

### Solution

Apply the Ratio Test:

L = lim (n→∞) |((-1)⁽ⁿ⁺¹⁾ x⁽ⁿ⁺¹⁾) / √(n+1) * √n / ((-1)ⁿ xⁿ)|

L = |x| lim (n→∞) (√n / √(n+1))

L = |x|

The series converges for **|x| < 1**, so the **radius of convergence** **r = 1**.

*All images were created with MATLAB.*