I'm currently looking through the Wikipedia Article about the ratio test for convergence of a series. The article includes a decision diagram for the ratio test.
The diagram look something like this:
Let's take a look at $\sum_{k = 1}^{\infty} a_n$ (where $a_n \in \mathbb{R} $ or $a_n \in \mathbb{C}$ for every $n \in \mathbb{N}$).
- $$\limsup_{n \rightarrow \infty} |\frac{a_{n+1}}{a_n}| < 1$$ Means the series $\sum_{k = 1}^{\infty} a_n$ converges absolutely.
- $$\lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_n}| > 1$$ Means the series $\sum_{k = 1}^{\infty} a_n$ diverges.
- $$|\frac{a_{n+1}}{a_n}| \geq 1 \hspace{15px} \text{(for almost all n} \in \mathbb{N})$$ Means the series $\sum_{k = 1}^{\infty} a_n$ diverges.
Now I have two questions regarding the decision diagram.
- Can't we simply combine the requirements of the last two decisions into $$\liminf_{n \rightarrow \infty} |\frac{a_{n+1}}{a_n}| \geq 1$$
- What are the requirements for the ratio test to fail (e.g. come to no conclusion)?
Counter-example: $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}n = \ln 2$$
(Convergence is provable by the alternating series test.)
But, $$\liminf_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \liminf_{n \to \infty} \frac n{n+1} = 1$$