Q. 67 - Calculus

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.

Step-by-Step Solution

Answer

Hence, proved.

Step 1. Given Information.

$a_k$ is a rational function of k.

Step 2. Ratio test.

$$\begin{gathered} The ratio test states that if\sum _{k=1}^{\infty }{a}_k be a series, if L = \underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}, then \\ i. If L<1 series converges. \\ ii. If L=1 the test is incconlusive. \\ iii. If L>1 series diverges. \end{gathered}$$

Step 3. General rational term.

$$\begin{gathered} Let r\left(x\right) = \sum _{i=0}^{\infty }\frac{a_i}{b_i}{x}^i, where b_i\ne 0. \\ The ratio \frac{r(x+1)}{r\left(x\right)} = \frac{\frac{a_i}{b_i}{\left(x+1\right)}^i}{\frac{a_i}{b_i}{x}^i} = \frac{{\left(x+1\right)}^i}{x^i} = {\left(1+\frac{1}{x}\right)}^i. \\ \underset{x\to \infty }{\lim }\frac{r(x+1)}{r\left(x\right)} = \underset{x\to \infty }{\lim } {\left(1+\frac{1}{x}\right)}^i = \underset{x\to \infty }{\lim } {\left(1+0\right)}^i = 1. \\ Thus the value of the ratio is 1. \\ Therefore the ratio test becomes inconclusive. \end{gathered}$$