Power Series and Interval of Convergence Calculator

Power series are fundamental in calculus and analysis, providing a way to represent functions as infinite sums of terms. The interval of convergence determines where these series converge to the original function. This calculator helps you determine the interval of convergence for any given power series.

What is a Power Series?

A power series is an infinite series of the form:

f(x) = Σ (from n=0 to ∞) aₙ(x - c)ⁿ = a₀ + a₁(x - c) + a₂(x - c)² + a₃(x - c)³ + ...

Where:

aₙ are coefficients
c is the center of the series
x is the variable

Power series are used to represent functions that can be differentiated or integrated term by term within their interval of convergence.

Interval of Convergence

The interval of convergence is the set of all x-values for which the power series converges. It can be determined using the Ratio Test or the Root Test.

The interval of convergence is typically expressed as (c - R, c + R), where R is the radius of convergence.

After finding the radius of convergence, you must check the endpoints separately to determine if the series converges at x = c - R and x = c + R.

How to Calculate Interval of Convergence

Identify the power series and its general term aₙ(x - c)ⁿ.
Apply the Ratio Test or Root Test to find the radius of convergence R.
Check for convergence at the endpoints x = c - R and x = c + R.
Combine the results to form the interval of convergence.

Using the Ratio Test:

lim (n→∞) |aₙ₊₁ / aₙ| = L

If L < 1, the series converges, and R = 1/L.

Examples of Power Series

Example 1: Geometric Series

Consider the series:

Σ (from n=0 to ∞) xⁿ = 1 + x + x² + x³ + ...

Using the Ratio Test:

lim (n→∞) |xⁿ⁺¹ / xⁿ| = |x|

The series converges when |x| < 1, so the radius of convergence is R = 1. Checking the endpoints:

At x = 1: The series becomes 1 + 1 + 1 + ... which diverges.
At x = -1: The series becomes 1 - 1 + 1 - 1 + ... which diverges.

Therefore, the interval of convergence is (-1, 1).

Example 2: Exponential Series

Consider the series:

Σ (from n=0 to ∞) xⁿ / n! = 1 + x + x²/2! + x³/3! + ...

Using the Ratio Test:

lim (n→∞) |(xⁿ⁺¹ / (n+1)!) / (xⁿ / n!)| = lim (n→∞) |x / (n+1)| = 0

Since the limit is 0 for all x, the series converges for all x. Therefore, the interval of convergence is (-∞, ∞).

FAQ

What is the difference between radius of convergence and interval of convergence?

The radius of convergence is the distance from the center c where the series converges. The interval of convergence includes the radius and checks the endpoints to determine if the series converges at those points.

How do I know if a power series converges at the endpoints?

You must check the endpoints separately using tests like the Ratio Test or Root Test. Sometimes the series may converge at one endpoint but not the other.

Can a power series converge for all x?

Yes, some power series have an infinite radius of convergence and converge for all real numbers. The exponential series is an example of this.