Question 3the,Test whether following sequence Convergent / divergent(6)5+3n²3"+2(P)n=2(7)

Name: Question 3the,Test whether following sequence Convergent / divergent(
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Description: Question 3the,Test whether following sequence Convergent / divergent(6)5+3n²3"+2(P)n=2(7)

a) Given that ∑n=1∞1n3Here f(x)=1n3 and p=3we know that, A p-series ∑ 1npconverges if…

Answered: bəs st whether following seq. uestion 3…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 49E

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Step 1

$a) G i v e n t h a t \sum_{n = 1}^{\infty} \frac{1}{n^{3}} H e r e f (x) = \frac{1}{n^{3}} a n d p = 3 w e k n o w t h a t, A p - s e r i e s \sum \frac{1}{n^{p}} c o n v e r g e s i f a n d o n l y i f p > 1 . B y p - s e r i e s t e s t, p = 3 > 1 \sum_{n = 1}^{\infty} \frac{1}{n^{3}} i s c o n v e r g e n t . b) G i v e n t h a t \sum_{n = 1}^{\infty} 5 {(- \frac{3}{4})}^{n - 1} H e r e a_{n} = 5 {(- \frac{3}{4})}^{n - 1} a_{n + 1} = 5 {(- \frac{3}{4})}^{n}$

Step 2

$N o w, \frac{a_{n + 1}}{a_{n}} = \frac{5 {(- \frac{3}{4})}^{n}}{5 {(- \frac{3}{4})}^{n - 1}} \frac{a_{n + 1}}{a_{n}} = \frac{5 {(- \frac{3}{4})}^{n}}{5 {(- \frac{3}{4})}^{n} {(- \frac{3}{4})}^{- 1}} \frac{a_{n + 1}}{a_{n}} = \frac{1}{{(- \frac{3}{4})}^{- 1}} \frac{a_{n + 1}}{a_{n}} = - \frac{3}{4} W e k n o w t h a t, I f \lim_{n \to \infty} \frac{a_{n + 1}}{a_{n}} = L a n d i f L < 1 t h e n t h e s e r i e s c o n v e r g e s b u t i f L > 1 t h e n t h e s e r i e s d i v e r g e s H e r e L = - \frac{3}{4} < 1 B y r a t i o t e s t, \sum_{n = 1}^{\infty} 5 {(- \frac{3}{4})}^{n - 1} i s c o n v e r g e s .$

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