Find Interval Of Convergence Calculator

Easily determine the radius and interval of convergence for a power series given the center and the limit L from the Ratio/Root Test.

Calculate Interval of Convergence

What is the Interval of Convergence?

The interval of convergence of a power series is the set of all x-values for which the series converges. A power series centered at 'a' has the form ∑ c_n(x-a)ⁿ, where c_n are the coefficients and 'a' is the center.

For any power series, there are three possibilities regarding its convergence:

1. The series converges only at its center x = a.
2. The series converges for all real numbers x.
3. There is a positive number R, called the radius of convergence, such that the series converges absolutely for |x – a| < R and diverges for |x - a| > R. The series may converge or diverge at the endpoints x = a – R and x = a + R.

The interval of convergence includes the open interval (a – R, a + R) and possibly one or both endpoints, depending on the series' behavior at x = a – R and x = a + R. Finding the interval of convergence is crucial for understanding where a power series representation of a function is valid.

This calculator is useful for students of calculus (typically Calculus II or III), engineers, and scientists who work with series expansions of functions.

A common misconception is that the Ratio Test or Root Test alone determines the full interval of convergence. While they find the radius R and the open interval (a-R, a+R), the convergence at the endpoints x=a-R and x=a+R must be tested separately using other convergence tests for series of constants.

Interval of Convergence Formula and Mathematical Explanation

To find the interval of convergence of a power series ∑ c_n(x-a)ⁿ, we typically follow these steps:

1. Apply the Ratio Test (or Root Test): We calculate the limit L:
   L = lim_n→∞ | (c_n+1(x-a)ⁿ⁺¹) / (c_n(x-a)ⁿ) | = lim_n→∞ |c_n+1/c_n| |x-a|
   or using the Root Test:
   L = lim_n→∞ |c_n(x-a)ⁿ|^1/n = lim_n→∞ |c_n|^1/n |x-a|
   Let L₀ = lim_n→∞ |c_n+1/c_n| or L₀ = lim_n→∞ |c_n|^1/n. Then the series converges if L₀|x-a| < 1.
2. Determine the Radius of Convergence (R):
   • If L₀ = 0, then L₀|x-a| = 0 < 1 for all x, so R = ∞.
   • If L₀ = ∞, then L₀|x-a| < 1 only if x=a, so R = 0.
   • If 0 < L₀ < ∞, then L₀|x-a| < 1 implies |x-a| < 1/L₀, so R = 1/L₀.
3. Find the Open Interval of Convergence: The series converges absolutely for |x-a| < R, which is the open interval (a-R, a+R). If R=∞, the interval is (-∞, ∞). If R=0, it's just {a}.
4. Test the Endpoints: If R is finite and positive, we must test the series for convergence at x = a – R and x = a + R by substituting these values into the original series and using tests like the p-series test, alternating series test, comparison test, etc.
5. State the Interval of Convergence: Combine the open interval with the behavior at the endpoints. For example, if it converges at x=a-R and diverges at x=a+R, the interval is [a-R, a+R).

Variables Table:

Variable	Meaning	Unit	Typical range
a	Center of the power series	Dimensionless	Any real number
cn	Coefficients of the power series	Varies	Varies
L0	Limit from Ratio/Root test on coefficients	Dimensionless	≥ 0 or ∞
R	Radius of convergence	Dimensionless	≥ 0 or ∞
x	Variable of the power series	Dimensionless	Real numbers

Table explaining the variables involved in finding the interval of convergence.

Practical Examples (Real-World Use Cases)

Example 1: Geometric Series

Consider the power series ∑ xⁿ (which is ∑ 1*(x-0)ⁿ, so a=0, c_n=1).

1. L₀ = lim_n→∞ |1/1| = 1.

2. R = 1/L₀ = 1/1 = 1.

3. Open interval: (0-1, 0+1) = (-1, 1).

4. Endpoints: x = -1 (∑ (-1)ⁿ, diverges) and x = 1 (∑ 1ⁿ, diverges).

5. Interval of convergence: (-1, 1).

Example 2: Alternating Series

Consider the power series ∑ ((-1)ⁿ * (x-2)ⁿ) / n (so a=2, c_n=(-1)ⁿ/n).

1. L₀ = lim_n→∞ | ((-1)ⁿ⁺¹/(n+1)) / ((-1)ⁿ/n) | = lim_n→∞ n/(n+1) = 1.

2. R = 1/L₀ = 1/1 = 1.

3. Open interval: (2-1, 2+1) = (1, 3).

4. Endpoints:
x = 1: ∑ ((-1)ⁿ * (-1)ⁿ) / n = ∑ 1/n (Harmonic series, diverges).
x = 3: ∑ ((-1)ⁿ * (1)ⁿ) / n = ∑ (-1)ⁿ/n (Alternating harmonic series, converges conditionally).

5. Interval of convergence: (1, 3].

How to Use This Interval of Convergence Calculator

1. Enter the Center (a): Input the value of 'a', the center of your power series ∑c_n(x-a)ⁿ.
2. Enter the Limit (L): Determine L₀ = lim |c_n+1/c_n| (or from the root test) based on the coefficients c_n of your series. Enter this value. If the limit is infinity, type "infinity"; if it's 0, type "0".
3. Specify Endpoint Behavior: If R is finite and positive (L is finite and positive), you need to manually check convergence at x=a-R and x=a+R. Select whether the series "Converges Absolutely," "Converges Conditionally," or "Diverges" at each endpoint using the dropdowns. These fields are enabled only when L>0 and finite.
4. Calculate: The results update automatically, or you can click "Calculate".
5. Read Results: The calculator will show the Radius of Convergence (R), the open interval (a-R, a+R), and the final interval of convergence based on your endpoint inputs.
6. Visualize: The chart shows the interval on a number line.

Understanding the interval of convergence tells you the range of x-values for which the power series provides a valid representation or approximation of a function.

Key Factors That Affect Interval of Convergence Results

• Coefficients (c_n): The nature of the coefficients c_n directly influences the limit L₀ and thus the radius R. Faster growing coefficients (relative to n) tend to result in a smaller R, while slower growing or decaying coefficients can lead to a larger R or even R=∞.
• Center (a): The center 'a' shifts the interval along the x-axis but does not change its width (2R). The interval is always centered at 'a'.
• Limit L₀: The value of L₀ is inversely related to R (when L₀ is positive and finite). A larger L₀ means a smaller R, and vice versa.
• Behavior at Endpoints: Whether the series converges or diverges at x=a-R and x=a+R determines if the interval is open, closed, or half-open. This often depends on subtle properties of c_n when combined with (±R)ⁿ.
• Type of Series at Endpoints: When x is an endpoint, the power series becomes a series of constants. Its convergence depends on whether it's a p-series, alternating series, geometric series, etc., at those points.
• Absolute vs. Conditional Convergence: At the endpoints, the series might converge absolutely or conditionally, affecting how the interval is written and understood (though the calculator groups them under "Converges" for interval notation, we distinguish them in the dropdown). Knowing the type of convergence can be important in further analysis.

Frequently Asked Questions (FAQ)

What is a power series?

A power series is an infinite series of the form ∑ c_n(x-a)ⁿ, where c_n are coefficients, 'x' is a variable, and 'a' is the center.

What is the radius of convergence?

The radius of convergence, R, is a non-negative number or ∞ such that the power series converges absolutely for |x-a| < R and diverges for |x-a| > R.

How do I find the limit L₀?

You use the Ratio Test (L₀ = lim |c_n+1/c_n|) or the Root Test (L₀ = lim |c_n|^1/n) applied to the coefficients of the power series.

What if R=0?

If R=0, the interval of convergence is just the single point {a}.

What if R=∞?

If R=∞, the interval of convergence is (-∞, ∞), meaning the series converges for all real x.

Why do I need to test endpoints separately?

The Ratio and Root Tests are inconclusive when the limit of the ratio (or root) is exactly 1, which corresponds to the endpoints |x-a| = R. You need other tests (like comparison, integral, alternating series test) for these specific x-values.

Can the interval of convergence be empty?

No, at the very least, a power series always converges at its center x=a.

What's the difference between absolute and conditional convergence at an endpoint?

If the series ∑ |c_n(±R)ⁿ| converges, the series converges absolutely at that endpoint. If ∑ c_n(±R)ⁿ converges but ∑ |c_n(±R)ⁿ| diverges, it converges conditionally.