ratio test

Consider the series $\displaystyle\sum_{n=1}^\infty a_n$.

• If $\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| < 1$, then $\displaystyle\sum_{n=1}^\infty a_n$ is absolutely convergent.
• If $\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| > 1$, then $\displaystyle\sum_{n=1}^\infty a_n$ is divergent.
• If $\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = 1$, then the ratio test is inconclusive.

root test

Consider the series $\displaystyle\sum_{n=1}^\infty a_n$.

• If $\displaystyle\lim_{n\to\infty}\sqrt[n]{|a_n|} < 1$, then $\displaystyle\sum_{n=1}^\infty a_n$ is absolutely convergent.
• If $\displaystyle\lim_{n\to\infty}\sqrt[n]{|a_n|} > 1$, then $\displaystyle\sum_{n=1}^\infty a_n$ is divergent.
• If $\displaystyle\lim_{n\to\infty}\sqrt[n]{|a_n|} = 1$, then the root test is inconclusive.

an example

<aside>

Determine if $\displaystyle\sum_{n=1}^\infty \frac{1}{2^n}$ is convergent or divergent using the ratio and root tests.

</aside>

We know that this series is convergent because it’s a geometric series with common ratio $\frac{1}{2}$, but let’s use the ratio and root tests to determine this.

Root test:

$$ \lim_{n\to\infty}\sqrt[n]{|a_n|}=\sqrt[n]{\left|\frac{1}{2^n}\right|} = \lim_{n\to\infty}\frac{1}{2} = \frac{1}{2} $$

Ratio test:

$$ \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty}\left|\frac{(1/2)^{n+1}}{(1/2)^n}\right| = \lim_{n\to\infty}\frac{1}{2} = \frac{1}{2} $$

<aside>

Is the series $\displaystyle\sum_{n=1}^\infty \frac{(-10)^n}{n!}$ convergent or divergent?

</aside>

We can use the ratio test here:

$$ \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty}\left|\frac{(-10)^{n+1}/(n+1)!}{(-10)^n/n!}\right| = \lim_{n\to\infty}\left|\frac{-10}{n+1}\right| = \lim_{n\to\infty}\frac{10}{n+1} = 0 < 1 $$