Ratio_Test

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Definition

Let $(a_n)$ be a sequence of positive terms.

• If $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = r < 1$, then the series $\sum_{n=1}^{\infty} a_n$ converges.
• If $r > 1$ (or the limit is $\infty$), the series diverges.
• If $r = 1$, the test is inconclusive.

Proof

If $r < 1$, choose $r < s < 1$. For large $n$, $\frac{a_{n+1}}{a_n} < s$.

Thus $a_{n+1} < s \cdot a_n < s^2 \cdot a_{n-1} < \cdots$, so $a_n < Cs^n$ for some constant $C$.

Since $\sum s^n$ converges (geometric series with $|s| < 1$), $\sum a_n$ converges by the Comparison_Test.