Question

Prove that the ratio test will be inconclusive on every series of the form $\sum _{k=1}^{\infty }{a}_{k}where{a}_k$is a rational function of k.

Short answer

Hence, proved.

Step-by-step solution

Step 1. Given Information.

$a_k$ is a rational function of k.

Step 2. Ratio test.

$$\begin{gathered} Theratioteststatesthatif\sum _{k=1}^{\infty }{a}_{k}beaseries,ifL=\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k},then \\ i.IfL<1seriesconverges. \\ ii.IfL=1thetestisincconlusive. \\ iii.IfL>1seriesdiverges. \end{gathered}$$

Step 3. General rational term.

$$\begin{gathered} Letr\left(x\right)=\sum _{i=0}^{\infty }\frac{a_i}{b_i}{x}^i,where{b}_i\ne 0. \\ Theratio\frac{r(x+1)}{r\left(x\right)}=\frac{\frac{a_i}{b_i}{\left(x+1\right)}^i}{\frac{a_i}{b_i}{x}^i}=\frac{{\left(x+1\right)}^i}{x^i}={\left(1+\frac{1}{x}\right)}^i. \\ \underset{x\to \infty }{\lim }\frac{r(x+1)}{r\left(x\right)}=\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^i=\underset{x\to \infty }{\lim }{\left(1+0\right)}^i=1. \\ Thusthevalueoftheratiois1. \\ Thereforetheratiotestbecomesinconclusive. \end{gathered}$$