Question

Fill in the blanks to complete each of the following theorem statements.

Tests for Monotonicity: A sequence {ak} is increasing if it passes any of the following tests:

The Ratio Test: All terms are positive and $\_\_\_\_\ge 1\mathrm{for}\mathrm{all}k\ge 1$

Short answer

The required answer is$\frac{a_{k+1}}{a_k}\ge 1\mathrm{for}\mathrm{all}k\ge 1$

Step-by-step solution

Step 1. Given Information

The given term is ratio test.

Step 2. Explanation

The ratio test follows from the fact that a fraction $\frac{a}{b}$is greater than or equal to 1 exactly when the numerator is greater than or equal to denominator.

Thus,$\frac{a_{k+1}}{a_k}\ge 1\mathrm{for}\mathrm{all}k\ge 1$