State the following. The Ratio Test

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$$\textbf{RATIO TEST}$$

Suppose we have the series $\sum a_n$

Now consider the Limit $L = \lim\limits_{n \to \infty}\left|\dfrac{a_{n+1}}{a_n} \right|$

If $L < 1$, the series is absolutely convergent

If $L > 1$, the series is divergent

If $L = 1$, this test is not useful, try something else}

The Ratio Test states that: (i) If $\lim_{n \rightarrow \infty}|a_{n+1}/a_{n}| < 1$, then $\sum a_n$ converges (absolutely). (ii) If $\lim_{n \rightarrow \infty}|a_{n+1}/a_{n}| > 1$, then $\sum a_n$ diverges. (iii) If $\lim_{n \rightarrow \infty}|a_{n+1}/a_{n}| = 1$, then the Test is inconclusive.

$\textbf{The Ratio Test:}$

For a series $\sum a_n$, consider the following limit:

$$\begin{align*} \lim_{n\to\infty}\left| \frac{a_{n+1}}{a_n} \right| = L. \end{align*}$$

$$\begin{align*} (i) \quad &\quad \quad \text{If } L < 1, \text{ then the series is absolutely convergent.} \\ (ii) \quad &\quad \quad \text{If } L > 1, \text{ then the series diverges.} \\ (iii) \quad &\quad \quad \text{If } L = 1, \text{ then this test is inconclusive.} \end{align*}$$