Q. 6 - Calculus

Question

Use Exercise 5 to explain why the ratio test will be inconclusive for every series $\sum _{k=1}^{\infty }{a}_k$ in which $a_k$ is a rational function of $k$ .

Step-by-Step Solution

Answer

It is explained.

Step 1. Given Information.

The given series is $\sum _{k=1}^{\infty }{a}_k$.

Step 2. Explanation.

Let $$\begin{gathered} r\left(x\right)=\frac{a_0}{b_0}x+\frac{a_1}{b_1}x+\frac{a_2}{b_2}{x}^2+\dots +\frac{a_n}{b_n}{x}^n+\dots \text{ with }{b}_i\ne 0 \\ =r\left(x\right)=\sum _{i=0}^{\infty }\frac{a_i}{b_i}{x}^i \\ r(x+1)=\sum _{i=0}^{\infty }\frac{a_i}{b_i}(x+1{)}^i \\ \underset{x\to \infty }{\lim }\frac{r(x+1)}{r\left(x\right)}=\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^{\text{'}} \\ =1 \end{gathered}$$

That's why the ratio test becomes inconclusive for the series $\sum _{k=1}^{\infty }{a}_k$ in which $a_k$ is the rational function of $k$.