Radius of Convergence Calculator - Power Series Tool

Radius of Convergence Calculator

For a power series Σ cn(x-a)n, find the Radius of Convergence R using the limit L from the Ratio Test.

You need to calculate this limit first. Enter a non-negative number or 'Infinity'.
The value 'a' in (x-a)n. Often 0 for Maclaurin series.

Radius of Convergence (R):

Interval of Convergence:

(Endpoints require separate testing)

What is the Radius of Convergence?

For a given power series centered at x = a, written in the form:

∑∞n=0   cn (x - a)n = c₀ + c₁(x-a) + c₂(x-a)² + c₃(x-a)³ + ...

The Radius of Convergence (R) is a non-negative number (or infinity) that describes how "far" from the center a the series converges (i.e., sums to a finite value). Specifically:

  • The series converges absolutely for all x such that |x - a| < R.
  • The series diverges for all x such that |x - a| > R.

The set of all x-values for which the series converges is called the Interval of Convergence. If R is finite and positive, this interval is at least (a - R, a + R). The convergence at the endpoints (x = a - R and x = a + R) must be tested separately.

  • If R = 0, the series converges only at the center x = a.
  • If R = ∞, the series converges for all real numbers x.

How This Radius of Convergence Calculator Works

This calculator finds the Radius of Convergence (R) using the result from the Ratio Test applied to the coefficients of the power series. The Ratio Test is a common method for determining R.

The Ratio Test involves calculating the limit:

L = limn→∞   | an+1 / an |

Where an = cn(x-a)n is the n-th term of the series. Applying this leads to needing the limit related to the coefficients:

Lcoeff = limn→∞   | cn+1 / cn |

Important Limitation: Calculating symbolic limits automatically is computationally complex. Therefore, this calculator requires YOU to calculate the limit L = Lcoeff first and enter its value.

  1. Calculate the Limit (L): Using calculus techniques (often involving simplifying the ratio | cn+1 / cn | and finding its limit as n approaches infinity), determine the value of L.
  2. Input Limit L: Enter the calculated value of L into the first field. This must be a non-negative number (0, a positive value, or you can type "Infinity").
  3. Input Center (a): Enter the center point a of the power series. This is often 0 for Maclaurin series.
  4. Click Calculate: Press the "Calculate Radius (R)" button.
  5. Calculate R: The calculator determines the Radius of Convergence R based on the value of L you provided:
    • If L = 0, then R = ∞ (infinity).
    • If 0 < L < ∞ (L is a finite positive number), then R = 1 / L.
    • If L = ∞ (infinity), then R = 0.
  6. Determine Interval: Based on R and a, the calculator determines the open interval (a - R, a + R).
  7. Display Results: The calculator shows the calculated Radius R (using the ∞ symbol if applicable) and the open Interval of Convergence. It reminds you that endpoint testing is separate.

Frequently Asked Questions (FAQs)

Q1: Why do I have to calculate the limit L myself?

Finding the limit L = lim |cn+1 / cn| often involves algebraic simplification (dealing with factorials, powers of n, etc.) and applying limit rules from calculus. Performing this symbolic manipulation and limit evaluation automatically requires a sophisticated Computer Algebra System (CAS), which is beyond the scope of a simple browser-based JavaScript calculator. This tool focuses on the final step: calculating R *from* the limit L.

Q2: What is the Ratio Test?

The Ratio Test is a convergence test for infinite series. For a series Σan, you calculate L = lim |an+1 / an|.

  • If L < 1, the series converges absolutely.
  • If L > 1 (or L = ∞), the series diverges.
  • If L = 1, the test is inconclusive.

When applied to power series Σ cn(x-a)n, the test shows convergence when lim |cn+1(x-a)n+1 / (cn(x-a)n)| < 1, which simplifies to |x-a| * lim |cn+1 / cn| < 1, or |x-a| * L < 1. This leads directly to the condition |x-a| < 1/L, defining R = 1/L.

Q3: What is the Interval of Convergence?

It's the complete set of x values for which the power series converges. The Radius of Convergence R gives you the "width" of this interval around the center a. The interval always includes (a - R, a + R). To find the full interval, you must also test whether the series converges at the endpoints x = a - R and x = a + R (using other convergence tests like the p-series test, alternating series test, etc.). This calculator only provides the open interval based on R.

Q4: What if the limit L is 0 or Infinity?

  • If L = 0: The condition |x-a| * 0 < 1 is true for all x. This means the series converges everywhere, and the Radius of Convergence is R = ∞. The interval is (-∞, +∞).
  • If L = ∞: The condition |x-a| * ∞ < 1 is only true if |x-a| = 0, which means x = a. The series converges only at its center, and the Radius of Convergence is R = 0. The interval is just the single point {a}.

This calculator handles these cases correctly.

Q5: What about the Root Test?

The Root Test is another method to find the radius of convergence. It involves calculating L = lim |cn|1/n. The radius is still given by R = 1/L (with the same interpretations for L=0 and L=∞). If you calculated L using the Root Test, you can still enter that value into this calculator to find R.

Q6: Does the center 'a' affect the Radius R?

No. When using the Ratio Test or Root Test limit L based *only* on the coefficients cn, the radius R depends solely on L (R = 1/L). The center a determines *where* the interval of convergence is centered: (a - R, a + R), but not its width (which is 2R).