Ratio Test

Definition

The ratio test determines convergence of an infinite series ${\sum }_{n=0}^{\infty }{a}_n$ by examining the limit of successive term ratios:

$$L=\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|.$$

Intuition

If the ratio of successive terms approaches a constant $L$, the series eventually behaves like a geometric series with ratio $L$. Convergence follows when $L<1$.

Formal Description

The geometric series $\sum x^n$ converges iff $|x|<1$, with ratio of successive terms equal to $x$. By comparison:

• $L<1$: series converges (absolutely).
• $L>1$: series diverges.
• $L=1$: indeterminate — the test is inconclusive.

Key Results

The ratio test is especially effective when $a_n$ involves factorials or exponentials.

Example. For $\sum _{n=0}^{\infty }\frac{1}{n!}$:

$$\underset{n\to \infty }{\lim }\frac{\frac{1}{(n+1)!}}{\frac{1}{n!}}=\underset{n\to \infty }{\lim }\frac{1}{n+1}=0<1.$$

The series converges (its sum is $e$).

Applications

The ratio test is used to find the radius of convergence of a power series:

$$\left|x\right|\underset{n\to \infty }{\lim }\left|\frac{c_{n+1}}{c_n}\right|<1⟹R=\underset{n\to \infty }{\lim }\left|\frac{c_n}{c_{n+1}}\right|.$$

Trade-offs

The test is inconclusive when $L=1$: both the convergent p-series $\sum 1/n^2$ and the divergent harmonic series $\sum 1/n$ give $L=1$. Polynomial or rational $a_n$ always yield $L=1$; comparison or integral tests are needed instead.

Links

• Geometric Series
• Power Series
• Sequences and Series