Radius and Interval of Convergence Calculator Symbolab

This calculator helps you determine the radius and interval of convergence for power series. Power series are infinite sums of terms that often appear in calculus and analysis. Understanding convergence is crucial for determining where these series are valid and useful.

What is Convergence?

A power series is an infinite sum of terms of the form \( a_n(x - c)^n \), where \( a_n \) are coefficients and \( c \) is the center of the series. The radius of convergence is the distance from the center \( c \) where the series converges. The interval of convergence is the set of all \( x \) values where the series converges.

There are three possible scenarios for the interval of convergence:

The series converges only at \( x = c \) (radius = 0).
The series converges for all real numbers (radius = ∞).
The series converges for \( x \) in the interval \( (c - R, c + R) \), where \( R \) is the radius of convergence.

How to Calculate Radius and Interval

To find the radius of convergence, we use the ratio test or the root test. The ratio test is often simpler and involves taking the limit of the absolute value of the ratio of consecutive terms as \( n \) approaches infinity.

The ratio test states that if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \), then the series converges absolutely if \( L < 1 \) and diverges if \( L > 1 \). If \( L = 1 \), the test is inconclusive.

Once the radius \( R \) is found, the interval of convergence is determined by checking the endpoints \( c - R \) and \( c + R \).

The Formula

For a power series \( \sum_{n=0}^{\infty} a_n (x - c)^n \), the radius of convergence \( R \) is given by:

\( R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}} \)

Or using the ratio test:

\( R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right| \)

The interval of convergence is \( (c - R, c + R) \), possibly including one or both endpoints depending on the behavior at the boundaries.

Worked Example

Consider the series \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \). Let's find its radius and interval of convergence.

Apply the ratio test:
\( \lim_{n \to \infty} \left| \frac{x^{n+1}/(n+1)!}{x^n/n!} \right| = \lim_{n \to \infty} \left| \frac{x}{n+1} \right| = 0 \)
Since the limit is 0 for all \( x \), the series converges for all real numbers.
Therefore, the radius of convergence is \( R = \infty \), and the interval of convergence is \( (-\infty, \infty) \).

Interpreting Results

The radius of convergence tells you how far from the center \( c \) the series remains valid. A finite radius means the series converges within a certain distance from \( c \). An infinite radius means the series converges everywhere.

At the endpoints of the interval, you must check convergence separately. The series might converge at one endpoint, diverge at the other, or both.

FAQ

What if the ratio test gives L = 1?: The ratio test is inconclusive when \( L = 1 \). You may need to use another test or analyze the series differently.
Can the radius of convergence be negative?: No, the radius of convergence is always a non-negative real number. It represents a distance.
How do I know if the series converges at the endpoints?: You must check convergence separately at \( x = c - R \) and \( x = c + R \) using other tests like the nth-term test or comparison test.