The Ratio Test for Positive Series of Real Numbers

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:
\begin{align} \quad \frac{a_{n+1}}{a_n} \leq r \end{align}
  • So $a_{n+1} \leq ra_n$. We see that:
\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:
\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:
\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:

\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:

\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.

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