Ratio test

Definition

The Ratio Test is used to determine the convergence or divergence of an infinite series by examining the limit of the ratio of successive terms. It is particularly useful for series with factorials or exponential functions.

5 Must Know Facts For Your Next Test

1. The Ratio Test involves calculating $\lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} \right|$.
2. If the limit $L < 1$, the series converges absolutely.
3. If the limit $L > 1$ or is infinite, the series diverges.
4. If the limit $L = 1$, the Ratio Test is inconclusive, and other methods must be used to determine convergence.
5. The Ratio Test can simplify complex series, especially those involving factorials or powers.

Review Questions

• What does it mean if $\lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} \right| = L$ and $L < 1$?
• Describe a scenario where the Ratio Test would be inconclusive.
• Why is the Ratio Test particularly useful for series with factorials or exponentials?

Related terms

Root Test: A convergence test that examines the n-th root of the absolute value of terms in a series.

Convergence: A property of a series where its partial sums approach a finite limit as more terms are added.

Divergence: A property of a series where its partial sums do not approach a finite limit as more terms are added.