Interval of Convergence Calculator – Find Power Series Range

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Interval of Convergence Calculator

Find the radius and interval of convergence for power series with step-by-step solutions

Results
Enter a power series above and click “Calculate Convergence” to see the radius and interval of convergence.

Common Power Series Examples

Geometric Series

∑(n=0 to ∞) x^n

Radius: R = 1

Interval: (-1, 1)

Converges for |x| < 1, diverges at endpoints

Exponential Series

∑(n=0 to ∞) x^n/n!

Radius: R = ∞

Interval: (-∞, ∞)

Converges for all real values of x

Logarithmic Series

∑(n=1 to ∞) (-1)^(n+1) x^n/n

Radius: R = 1

Interval: (-1, 1]

Converges at x = 1, diverges at x = -1

Binomial Series

∑(n=0 to ∞) (α choose n) x^n

Radius: R = 1

Interval: Depends on α

Endpoint convergence varies with parameter α

Key Concepts and Definitions

Power Series

A power series is an infinite series of the form:

∑(n=0 to ∞) aₙ(x-c)^n

Where aₙ are coefficients, c is the center, and x is the variable. Power series are fundamental in mathematical analysis and have applications in physics, engineering, and other fields.

Radius of Convergence

The radius of convergence R is the distance from the center within which the series converges absolutely:

|x – c| < R

For |x – c| > R, the series diverges. At |x – c| = R, convergence depends on the specific series and requires endpoint testing.

Interval of Convergence

The interval of convergence is the set of all x-values for which the power series converges. It has the form:

(c – R, c + R), [c – R, c + R], or mixed

The exact form depends on whether the series converges at the endpoints x = c ± R, which must be tested separately.

Ratio Test

The ratio test is commonly used to find the radius of convergence:

L = lim(n→∞) |aₙ₊₁/aₙ|

If L exists, then R = 1/L. If L = 0, then R = ∞. If L = ∞, then R = 0. This test is particularly effective for series with factorials or exponentials.

Root Test

The root test provides another method for finding convergence:

L = lim(n→∞) ∜|aₙ|

Similar to the ratio test, R = 1/L when L exists. The root test is often useful when the coefficients involve nth powers.

Endpoint Testing

After finding the radius R, test convergence at x = c ± R using:

Alternating Series Test, p-Series Test, Comparison Test

The series may converge at one, both, or neither endpoint. This determines whether the interval uses parentheses (open) or brackets (closed).

Step-by-Step Solution Process

  1. Identify the Power Series
    Write the series in standard form ∑aₙ(x-c)^n and identify the coefficients aₙ and center c.
  2. Apply Convergence Test
    Use the ratio test or root test to find the limit L. Calculate R = 1/L (or R = ∞ if L = 0).
  3. Determine Basic Interval
    The series converges for |x – c| < R, giving the open interval (c - R, c + R).
  4. Test Left Endpoint
    Substitute x = c – R into the original series and test for convergence using appropriate tests.
  5. Test Right Endpoint
    Substitute x = c + R into the original series and test for convergence using appropriate tests.
  6. Write Final Answer
    Combine results to write the complete interval of convergence, using brackets for included endpoints.

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