Q. 67 Prove that the ratio test will b... [FREE SOLUTION]

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Chapter 7: Q. 67 (page 641)

Prove that the ratio test will be inconclusive on every series of the form ${鈭�}_{k = 1}^{鈭�} a_{k} w h e r e a_{k}$ 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.

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

Step 3. General rational term.

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

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91影视

Prove that the ratio test will be inconclusive on every series of the form ${鈭�}_{k = 1}^{鈭�} a_{k} w h e r e a_{k}$ 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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91影视

Prove that the ratio test will be inconclusive on every series of the form 鈭�k=1鈭�akwhereakis a rational function of k.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Ratio test.

Step 3. General rational term.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

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Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

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