# Partial sum

The sum of the first n terms of a series. The infinite series equals the limit of its partial sums, if that limit exists.

Maths (university) · Part 07: Sequences and series · Card 2 of 9 · Last updated 2026-10-19

## Analogy

A running total on a till receipt: the final bill is what the running total settles on.

## Example

**1 + 1/2 + 1/4 + 1/8 + ...**

S_1 = 1. S_2 = 1.5.
S_3 = 1.75. S_4 = 1.875.
Totals creep towards 2.
So the series sums to 2.

## In practice

Defines what it means for an infinite sum to have a value.

## Practice

### Flashcards

- **What is a partial sum?** The sum of the first n terms of a series.

### Interview questions

**Find the sum of 3 + 1.5 + 0.75 + ... and explain why it is valid.**

- Geometric with a = 3, r = 0.5
- |r| < 1 so the series converges
- Sum = a/(1 - r) = 3/0.5 = 6
- Partial sums approach 6

## Questions people ask

### How can infinitely many positive numbers add to a finite total?

If the terms shrink quickly enough, the running total stays below a ceiling. Adding 1, then 1/2, then 1/4 never passes 2. Each new term fills half of the remaining gap, so the total gets as close to 2 as you like.

### What does it mean if the partial sums diverge?

It means the running total never settles on one number. It may grow without bound, like 1 + 2 + 3 and so on, or oscillate, like 1 - 1 + 1 - 1. In either case the series has no sum.

## Related concepts

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

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