Elementary Differential Equations 8th edition by Boyce ch05

Elementary Differential Equations, with ODE Architect CD

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—————————————————————————— —— CHAPTER 5. ________________________________________________________________________ page 169 Chapter Five Section 5.1 1. Apply the ratio test : lim lim 8Ä_ 8" 8 ¸¸ ab kk B$ œ B$ œ B$Þ Hence the series converges absolutely for . The radius of convergence is B$ " 3 œ" Bœ# Bœ% . The series diverges for and , since the n-th term does not approach zero. 3. Applying the ratio test, lim lim #8# # #8 8x B B 8" xB 8" œœ ! Þ The series converges absolutely for all values of . Thus the radius of convergence is B 3 œ_ . 4. Apply the ratio test : lim lim 8" 8" 88 #B œ# B œ # B Þ Hence the series converges absolutely for , or . T # B B  "Î# he radius of convergence is . term does not approach 3 œ "Î# The series diverges for , since the B œ „"Î# n-th zero. 6. Applying the ratio test, lim lim 8" 8 a b 8BB 8" BB œ BB œ BB Þ 8 ! ! !! Hence the series converges absolutely for . The radius of convergence is " ! 3 BœB " . At , we obtain the At the other ! harmonic series divergent , which is . endpoint, , we obtain BœB " ! " 8œ" _ 8 " 8 , which is conditionally convergent. 7. Apply the ratio test :
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—————————————————————————— —— CHAPTER 5. ________________________________________________________________________ page 170 lim lim 8Ä_ 8 #8 " 8" # # 8 # ¸¸ ab kk $8 " B # $8 B # $ 8 $ œ B# œ B# Þ 8" " Hence the series converges absolutely for , or . T " $ B# " B# $ he radius of convergence is . 3 œ$ At and , the series diverges, since the Bœ & Bœ " n-th term does not approach zero. 8. Applying the ratio test, lim lim 88 " 8 8" 8 8 88 " x B 8 " 8xB œB œ B / , since lim lim 8 8 8 " 8" 8 œ" œ / Œ . Hence the series converges absolutely for . T B/ he radius of convergence is . 3 œ/ At , the series , since the Bœ„/ diverges n-th term does not approach zero. This follows from the fact that lim 8 8 8x / 8# 8 . È 1 10. We have , with , for . Therefore . 0 B œ / 0 B œ / 8 œ "ß #ß â 0 ! œ " B8B 8 Hence the Taylor expansion about is Bœ! ! B 8x B 8œ! _ 8 " . Applying the ratio test, lim lim 8" 8 8xB " 8" xB œ ! . The radius of convergence is . 3 œ_ 11. We have , with and , for . Clearly, 0 B œB 0 B œ" 0 B œ! 8œ# ßâ w8 0 " œ" and , with all other derivatives equal to . Hence the Taylor w zero expansion about is Bœ" ! Bœ" B"Þ Since the series has only a finite number of terms, the converges absolutely for all . B 14. We have , , , 0 B œ "Î "  B 0 B œ  "Î "  B 0 B œ #Î "  B â a b a b a b ww w #$ with , for . It follows that 0B œ " 8 x Î " B 8   " 0! œ " 8 x 8 8 8 8 a b aba b a b 8"
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—————————————————————————— —— CHAPTER 5. ________________________________________________________________________ page 171 for . Hence the Taylor expansion about is 8 ! B œ! ! " "B œ " B " ab 8œ! _ 8 8 . Applying the ratio test, lim lim 8Ä_ 8" 8 kk kk kk B B œB œ B . The series converges absolutely for , but diverges at .
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Elementary Differential Equations 8th edition by Boyce ch05...

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