---
tag: 9JQEWK5405
kind: statement
role: theorem
name: Kummer's test
msc: [40A05]
uses: [0CVVKY7GY2, 0NBRP6093V, 04Z5G0THN6, 0QNY6ZBXPN]
about: [FJKV01J9F1]
sources:
  - url: https://en.wikipedia.org/wiki/Ratio_test
---

## Statement

Let $(a_n)$ and $(c_n)$ be sequences of positive real numbers.
For each $n$ put
$$\rho_n = c_n \frac{a_n}{a_{n+1}} - c_{n+1}.$$

1. If there are $\delta > 0$ and $N$ with $\rho_n \ge \delta$ for all $n \ge N$, then $\sum a_n$ [[converges|0CVVKY7GY2]].
2. If $\sum 1/c_n$ diverges and there is $N$ with $\rho_n \le 0$ for all $n \ge N$, then $\sum a_n$ [[diverges|0NBRP6093V]].

## Remarks

For part 1, multiply $\rho_n \ge \delta$ by $a_{n+1}$ to get $c_n a_n - c_{n+1} a_{n+1} \ge \delta a_{n+1}$ for $n \ge N$.
Summing from $N$ to $m$ telescopes, so $\delta \sum_{n=N+1}^{m+1} a_n \le c_N a_N$, and the partial sums are bounded.
For part 2, the sequence $c_n a_n$ increases from $n = N$ on, so $a_n \ge c_N a_N / c_n$, and the [[comparison test|04Z5G0THN6]] with $\sum 1/c_n$ gives divergence.
The choice $c_n = 1$ gives the [[ratio test|0QNY6ZBXPN]], the choice $c_n = n$ gives [[Raabe's test|CD6CAQ6XDN]], and the choice $c_n = n \ln n$ gives [[Bertrand's test|6T3E3AXZT7]].

## Explanations

### L1 human

Kummer's test is a family of tests, one for each choice of comparison weights $c_n$.
It compares the decay of the terms $a_n$ with the growth of the weights.
The familiar ratio test is the simplest member of the family.

### L2 human

When the ratio test gives the limit $1$, one uses Kummer's test with weights that grow, such as $n$ or $n \ln n$.
Each such choice gives a finer test: Raabe's test catches the $p$-series, and Bertrand's test catches series like $\sum 1/(n (\ln n)^p)$.
The proof only uses telescoping sums and the comparison test.