Quotient Criterion

analysis

Definition

Quotient Criterion

The quotient criterion (or ratio test) is a convergence criterion for an infinite series ${\sum }_{n=0}^{\infty }{a}_n$:

1. If there exists $q<1$ such that $\left|\frac{a_{n+1}}{a_n}\right|\le q$ for almost all $n$, then the series is absolutely convergent.
2. If $\left|\frac{a_{n+1}}{a_n}\right|\ge 1$ for almost all $n$, then the series is divergent.

Limit Form

Using the limit superior and limit inferior:

• ${\mathrm{limsup}}_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|<1⟹$ absolutely convergent.
• ${\mathrm{liminf}}_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|>1⟹$ divergent.

Example

Let $a_n=\frac{n}{2^n}$. We examine the ratio:

$$\left|\frac{a\_n+1}{a_n}\right|=\frac{\frac{n+1}{2^{n+1}}}{\frac{n}{2^n}}=\frac{n+1}{2n}=\frac{1+\frac{1}{n}}{2}\to \frac{1}{2}$$

Since $\frac{1}{2}<1$, the series $\sum \frac{n}{2^n}$ converges.