Section 7: Ratio and Root Tests
The Ratio Test: Let $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$. If $L < 1$, then $\sum_{n=1}^\infty a_n$ converges absolutely. If $L > 1$, then $\sum_{n=1}^\infty a_n$ diverges. If $L = 1$, the test tells us nothing.
The ratio is particularly useful when we see factorials in the terms of the series.
Example: Does $\sum_{n=1}^\infty \dfrac{3^n}{n!}$ converge or diverge? Let's use the ratio test. $|a_{n+1}/a_n| = \frac{3^{n+1}}{(n+1)!} \cdot \frac{n!}{3^n} = \frac{3}{n+1} \to 0$ as $n \to \infty$. So $\sum_{n=1}^\infty \dfrac{3^n}{n!}$ converges.
Mini-Quiz: Is $\sum_{n=1}^\infty n\left(\frac{3}{5}\right)^{n+3}$ convergent or divergent?
The Root Test: Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. If $L < 1$, then $\sum_{n=1}^\infty a_n$ converges absolutely. If $L > 1$, then $\sum_{n=1}^\infty a_n$ diverges. If $L = 1$, the test tells us nothing.
Example: Let's test $\sum_{n=1}^\infty \left(1 + \frac{1}{n}\right)^{n^2}$ for convergence. We apply the root test. $\sqrt[n]{\left(1 + \frac{1}{n}\right)^{n^2}} = \left(1 + \frac{1}{n}\right)^n \to e > 1$ as $n \to \infty$. (To compute this limit, you can take logarithms and use L'Hospital's Rule.) So the series diverges.
Mini-Quiz: Does $\sum_{n=1}^\infty \left(\dfrac{en+\ln{2}}{\pi n + \sqrt{2}}\right)^n$ converge or diverge?