# p-series

The series of 1 / n^p converges if p > 1 and diverges if p <= 1. With p = 1 it is the harmonic series, which diverges slowly.

Maths (university) · Part 07: Sequences and series · Card 5 of 9 · Last updated 2026-10-19

## Analogy

A leaking tap: the water slows down fast enough (p > 1) or too slowly (p <= 1) to fill a finite bucket.

## Example

**Two p-series**

Sum of 1/n^2: p = 2, converges (to about 1.645).
Sum of 1/n: p = 1, diverges.
Sum of 1/sqrt(n): p = 0.5, diverges.
Sum of 1/n^3: converges.

## In practice

A benchmark to compare other series against.

## Practice

### Flashcards

- **For which p does the sum of 1/n^p converge?** p > 1.

## Questions people ask

### Why does the harmonic series diverge if its terms go to zero?

The terms shrink too slowly. Grouping them shows each block of terms from 1/3 to 1/4, then 1/5 to 1/8, and so on adds more than 1/2. Endless blocks of at least 1/2 push the total past any limit, even though each single term becomes tiny.

### How does the integral relate to p-series?

The integral test links the sum of 1 over n to the power p with the integral of 1 over x to the power p. We saw that integral converges exactly when p is above 1, so the series behaves the same way.

## 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)
- [Ratio test](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-6)

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