The Ratio Test for Positive Series of Real Numbers

We will now develop yet another important test for determining the convergence or divergence of a series. This test is known as the ratio test for positive series.

Theorem 1: Let $(a_n)_{n=1}^{\infty}$ be a positive sequence of real numbers and let $\displaystyle{\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho}$. a) If $0 \leq \rho < 1$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges. b) If $1 < \rho \leq \infty$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges. If $\rho = 1$ then this test is inconclusive.

• Proof of a): Suppose that $0 \leq \rho < 1$. Since $\displaystyle{\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho}$ for $r$ where $\rho < r < 1$ there exists an $N \in \mathbb{N}$ where if $n \geq N$ then:

(1)

\begin{align} \quad \frac{a_{n+1}}{a_n} \leq r \end{align}

• So $a_{n+1} \leq ra_n$. We see that:

(2)

\begin{align} \quad a_{N+1} & \leq r a_{N} \\ \quad a_{N+2} & \leq r a_{N+1} \leq r^2 a_{N} \\ \quad a_{N+3} & \leq r a_{N+2} \leq r^2 a_{N+1} \leq r^3 a_N \\ \quad & \vdots\\ \quad a_{N+k} & \leq \cdots \leq r^k a_N \end{align}

• So $\displaystyle{\sum_{n=N+1}^{\infty} a_n = \sum_{k=1}^{\infty} a_{N+k} \leq \sum_{k=1}^{\infty} r^k a_N}$. But the series $\displaystyle{\sum_{k=1}^{\infty} r^k a_N}$ converges as a geometric series since $0 \leq \rho < r < 1$, and by the comparison test we have that the subseries $\displaystyle{\sum_{n=N+1}^{\infty} a_n}$ converges also which implies that the whole series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.

• Proof of b) Suppose that $1 < \rho \leq \infty$. Since $\displaystyle{\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho}$, for $r$ such that $1 < r \rho$ we have that there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then:

(3)

\begin{align} \quad r \leq \frac{a_{n+1}}{a_n} \\ \end{align}

• So for $ra_n \leq a_{n+1}$. So for $n \geq N$ we have:

(4)

\begin{align} \quad ra_N & \leq a_{N+1} \\ \quad r^2a_N & \leq ra_{N+1} \leq a_{N+2} \\ \quad r^3a_N & \leq r^2 a_{N+1} \leq r^2 a_{N+2} \leq a_{N+3} \\ \quad \quad & \vdots \\ \quad r^ka_N & \leq \cdots \leq a_{N+k} \end{align}

• So $\displaystyle{\sum_{k=1}^{\infty} r^k a_N \leq \sum_{k=1}^{\infty} a_{N+k} = \sum_{n=N+1}^{\infty} a_n}$. Since $1 < r$ we have that the series $\displaystyle{\sum_{k=1}^{\infty} r^k a_N}$ diverges as a geometric series and by comparison the subseries $\displaystyle{\sum_{n=N+1}^{\infty} a_n}$ diverges so the whole series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.

If $\rho = 1$ then the series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ may converge or diverge. For example, consider the series $\displaystyle{\sum_{n=1}^{\infty} \frac{1}{n^2}}$. We know this series converges. Using the ratio test and we see that:

(5)

\begin{align} \quad \rho = \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{n^2}{(n+1)^2} = 1 \end{align}

We also know that the series $\displaystyle{\sum_{n=1}^{\infty} \frac{1}{n}}$ diverges, and using the ratio test we see that:

(6)

\begin{align} \quad \rho = \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{n}{n+1} = 1 \end{align}

So as you can see, if $\rho = 1$ then the ratio test gives us no information on the convergence/divergence of a series.