Ratio Test

The Ratio Test is a method to determine the convergence of series. It states that if the limit of the ratio of the $(n+1)$-th term to the $n$-th term is less than 1, then the sequence converges. If the limit is greater than 1, then the sequence diverges. If the limit is equal to 1, the test is inconclusive, in other words:

Be $(a_n) \subset \mathbb{R}^+$ a sequence of positive real numbers and exist $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L$. Then:

1. If $L < 1$, then the sequence $(a_n)$ converges.
2. If $L > 1$, then the sequence $(a_n)$ diverges.
3. If $L = 1$, then the test is inconclusive.

Let’s prove the Ratio Test:

Suppose that $L < 1$, then there exists $r \in \mathbb{R}$ such that $L < r < 1$. Since $L < r$, there exists $n_0 \in \mathbb{N}$ such that for all $n \in \mathbb{N}$ with $n \geq n_0$ we have that $\left| \frac{a_{n+1}}{a_n} \right| < r$. Then, we can say that:

As we know that $r < 1$, then:

Therefore the sequence $(a_n)$ converges.

If we operate the same way considering $L > 1$, let’s consider $r \in \mathbb{R}$ such that $1 < r < L$. Then, there exists $n_0 \in \mathbb{N}$ such that for all $n \in \mathbb{N}$ with $n \geq n_0$ we have that $\left| \frac{a_{n+1}}{a_n} \right| > r$. Then, we can say that:

As we know that $r > 1$, then:

Therefore the sequence $(a_n)$ diverges as it grows without bound.

Finally, if $L = 1$, the test is inconclusive and we can’t determine if the sequence converges or diverges.