Q. 0 Problem Zero: Read the section a... [FREE SOLUTION]

Chapter 7: Q. 0 (page 652)

Problem Zero: Read the section and make your own summary of the material.

Short Answer

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$1 . A l t e r n a t i n g s e r i e s t e s t : L e t {a_{k}} b e a s t r i c t l y d e c r e a \sin g s e q u e n c e o f p o s i t i v e n u m b e r s s u c h t h a t \lim_{k 鈫� 鈭�} a_{k} = 0 . T h e n t h e a l t e r n a t i n g s e r i e s {鈭�}_{k = 1}^{鈭�} {(- 1)}^{k + 1} a_{k} a n d {鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} a_{k} b o t h c o n v e r g e .$

$2 . A b s o l u t e C o n v e r g e n c e I m p l i e s C o n v e r g e n c e : I f t h e s e r i e s {鈭�}_{k = 1}^{鈭�} b_{k} c o n v e r g e s a b s o l u t e l y, t h e n i t c o n v e r g e s .$

$3 . R a t i o T e s t f o r A b s o l u t e C o n v e r g e n c e : L e t {鈭�}_{k = 1}^{鈭�} b_{k} b e a s e r i e s w i t h n o n z e r o t e r m s, a n d l e t 蚁 = \lim_{k 鈫� 鈭�} |\frac{b_{k + 1}}{b_{k}}| . i . I f 蚁 < 1, t h e s e r i e s c o n v e r g e s a b s o l u t e l y . i i . I f 蚁 > 1, t h e s e r i e s d i v e r g e s . i i i . I f 蚁 = 1, t h e t e s t i s i n c o n c l u s i v e .$

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3 . R a t i o T e s t f o r A b s o l u t e C o n v e r g e n c e : L e t {鈭�}_{k = 1}^{鈭�} b_{k} b e a s e r i e s w i t h n o n z e r o t e r m s, a n d l e t 蚁 = \lim_{k 鈫� 鈭�} |\frac{b_{k + 1}}{b_{k}}| . i . I f 蚁 < 1, t h e s e r i e s c o n v e r g e s a b s o l u t e l y . i i . I f 蚁 > 1, t h e s e r i e s d i v e r g e s . i i i . I f 蚁 = 1, t h e t e s t i s i n c o n c l u s i v e .

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Problem Zero: Read the section and make your own summary of the material.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Summary of the section.

One App. One Place for Learning.

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