Q. 20 - Calculus

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 $a_k<0.$

Step-by-Step Solution

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 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 $\left\{a_k\right\}$is increasing.