Q. 0 - Calculus | StudyQuestionHub

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Q. 0

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

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

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

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

1Step 1. Given Information.

Given is the text content from the book.

2Step 2. Summary of the section.

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

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

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 \sum_{k = 1}^{\infty} 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 ρ = \lim_{k \to \infty} |\frac{b_{k + 1}}{b_{k}}| . i . I f ρ < 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 ρ > 1, t h e s e r i e s d i v e r g e s . i i i . I f ρ = 1, t h e t e s t i s i n c o n c l u s i v e .

2 Thinking Back

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