Question

Explain why we require the terms of the sequence$a_k$to be positive when we use the ratio test from Theorem 7.6.

Short answer

Ans: Wille using the Ratio test, all the terms in the sequence should be positive, otherwise the inequality will not hold true.

Consider $3>2$ holds true, if the terms are negative $-3>-2$ will not hold true.

Hence, it is necessary the terms of the sequence should all be positive while using the Ratio Test.

Step-by-step solution

Step 1. Given information.

given,

Consider the sequence$\left\{a_k\right\}$

Step 2. The objective is to explain why every term of the sequence{ak} should be positive while using the Ratio Test.

According to Ratio Test, the sequence $\left\{a_k\right\}$is an increasing sequence, if $\frac{a_{k+1}}{a_k}\ge 1$for all $k\ge 1$provided all the terms in the sequence are positive.

Step 3. Thus,

Wille using the Ratio test, all the terms in the sequence should be positive, otherwise, the inequality will not hold true.

Consider $3>2$ holds true, if the terms are negative $-3>-2$ will not hold true.

Hence, it is necessary the terms of the sequence should all be positive while using the Ratio Test.