Question

State a variation of the ratio test from Theorem 7.6 that would allow you to use ratios to test a sequence $\left\{a_k\right\}$for monotonicity when each $a_k<0.$

Short answer

According to the ratio test for Monotonicity, If the sequence $a_k<0$for all $k\ge 1$and $\frac{a_{k+1}}{a_k}\ge 1$for all $k\ge 1$then the sequence $\left\{a_k\right\}$is increasing.

Step-by-step solution

Step 1. Given information.

The given theorem states the ratio test for Monotonicity.

Step 2. ratio test for Monotonicity.

According to the ratio test for Monotonicity, If the sequence $a_k<0$for all $x\ge 1$and $\frac{a_{k+1}}{a_k}\ge 1$for all $k\ge 1$then the sequence role="math" localid="1649179280411" $\left\{a_k\right\}$is increasing.