# Radius of convergence

A power series (a sum of terms a_n x^n) converges for x with |x| < R, where R is the radius of convergence. Test the ends separately.

Maths (university) · Part 07: Sequences and series · Card 8 of 9 · Last updated 2026-10-19

## Analogy

A radio signal: it is clear within a certain distance of the tower and lost beyond it.

## Example

**Series of x^n / 2^n**

Ratio of terms: |x| / 2.
Converges when |x|/2 < 1.
So |x| < 2.
Radius R = 2.

## In practice

Says for which x a power series gives a valid formula.

## Practice

### Flashcards

- **What is the radius of convergence of the series of x^n/2^n?** 2.

## Questions people ask

### How do I find the radius of convergence?

Apply the ratio test to the terms, including the powers of x. Require the limit to be less than 1, and solve for x. For x to the n over 2 to the n, the condition is absolute x over 2 below 1, giving R = 2.

### What happens at the endpoints of the interval?

The ratio test says nothing at the ends, so test them separately by substituting the endpoint values. For x equal to 2 in x to the n over 2 to the n, the series is 1 + 1 + 1 and so on, which diverges.

## Related concepts

- [Partial sum](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-2)
- [Geometric series](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-3)
- [Divergence test](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-4)
- [p-series](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-5)

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