---
tag: 04Z5G0THN6
kind: statement
role: theorem
name: Comparison test
aliases: [direct-comparison-test]
msc: [40A05]
uses: [0MX1HZBBXJ]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_comparison_test
    locator: For series
same_as:
  wikidata: Q1050206
  wikipedia: Direct comparison test
---

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences with $0 \le a_n \le b_n$ for all $n$.
If $\sum b_n$ converges, then $\sum a_n$ converges, and $\sum a_n \le \sum b_n$.
If $\sum a_n$ diverges, then $\sum b_n$ diverges.

## Remarks

The partial sums of $\sum a_n$ are at most those of $\sum b_n$. The result follows from [[a series of nonnegative terms converges iff its partial sums are bounded|0MX1HZBBXJ]].
It is enough that $0 \le a_n \le b_n$ for all $n$ larger than some $N$.
For example, $0 \le 1/(n^2 + 1) \le 1/n^2$, so $\sum 1/(n^2 + 1)$ converges.
Mathlib states it for a summable upper bound, in `Summable.of_nonneg_of_le`.

## Explanations

### L1 human

If a series of nonnegative terms is smaller term by term than a series with a finite sum, it has a finite sum too.
If a smaller series already has an infinite sum, the larger one has an infinite sum as well.