Chapter 7: Q. 67 (page 641)

Prove that the ratio test will be inconclusive on every series of the form $\sum_{k = 1}^{\infty} a_{k} w h e r e a_{k}$ is a rational function of k.

Short Answer

Expert verified

Hence, proved.

Step by step solution

Step 1. Given Information.

$a_{k}$ is a rational function of k.

Step 2. Ratio test.

$T h e r a t i o t e s t s t a t e s t h a t i f \sum_{k = 1}^{\infty} a_{k} b e a s e r i e s, i f L = \lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}}, t h e n i . I f L < 1 s e r i e s c o n v e r g e s . i i . I f L = 1 t h e t e s t i s i n c c o n l u s i v e . i i i . I f L > 1 s e r i e s d i v e r g e s .$

Step 3. General rational term.

$L e t r (x) = \sum_{i = 0}^{\infty} \frac{a_{i}}{b_{i}} x^{i}, w h e r e b_{i} \neq 0 . T h e r a t i o \frac{r (x + 1)}{r (x)} = \frac{\frac{a_{i}}{b_{i}} {(x + 1)}^{i}}{\frac{a_{i}}{b_{i}} x^{i}} = \frac{{(x + 1)}^{i}}{x^{i}} = {(1 + \frac{1}{x})}^{i} . \lim_{x \to \infty} \frac{r (x + 1)}{r (x)} = \lim_{x \to \infty} {(1 + \frac{1}{x})}^{i} = \lim_{x \to \infty} {(1 + 0)}^{i} = 1 . T h u s t h e v a l u e o f t h e r a t i o i s 1 . T h e r e f o r e t h e r a t i o t e s t b e c o m e s i n c o n c l u s i v e .$

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Prove that the ratio test will be inconclusive on every series of the form $\sum_{k = 1}^{\infty} a_{k} w h e r e a_{k}$ is a rational function of k.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Ratio test.

Step 3. General rational term.

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Prove that the ratio test will be inconclusive on every series of the form $\sum_{k = 1}^{\infty} a_{k} w h e r e a_{k}$ is a rational function of k.