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 8 Ä _ 8 Ä _ 8" 8 ¸ ¸ a b k k a b k k k k B  $ B  $ œ B  $ œ B  $ Þ Hence the series converges absolutely for . The radius of convergence is k k B  $  " 3 œ " B œ # B œ % . The series diverges for and , since the n-th term does not approach zero. 3. Applying the ratio test, lim lim 8 Ä _ 8 Ä _ #8# # #8 k k k k a b 8x B B 8  " x B 8  " œ œ ! 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 Ä _ 8 Ä _ 8" 8" 8 8 k k k k k k k k # B # B œ # B œ # B Þ Hence the series converges absolutely for , or . T # B B  "Î# k k k k 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 Ä _ 8 Ä _ 8" 8 ¸ ¸ a b k k a ba b k k k k a b a b 8 B  B 8  " B  B 8  " œ B  B œ B  B Þ 8 ! ! ! ! Hence the series converges absolutely for . The radius of convergence is k k a b B  B  " ! 3 œ " B œ B  " . At , we obtain the At the other ! harmonic series divergent , which is . endpoint, , we obtain B œ B  " ! " a b 8œ" _ 8  " 8 , which is conditionally convergent. 7. Apply the ratio test :
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—————————————————————————— —— CHAPTER 5. ________________________________________________________________________ page 170 lim lim 8 Ä _ 8 Ä _ 8 # 8" 8" # # 8 # ¸ ¸ a b a b k k a b a b k k k k a b a b $ 8  " B  # $ 8 B  # $ 8 $ œ B  # œ B  # Þ 8  " " Hence the series converges absolutely for , or . T " $ k k k k B  #  " B  #  $ he radius of convergence is . 3 œ $ At and , the series diverges, since the B œ  & B œ  " n-th term does not approach zero. 8. Applying the ratio test, lim lim 8 Ä _ 8 Ä _ 8 8" 8 8" 8 8 k k a b ¸ ¸ a b a b k k k k 8 8  " x B 8 " 8  " 8x B œ B œ B 8  " / , since lim lim 8 Ä _ 8 Ä _ 8 8  8 " 8 " 8  " 8 œ "  œ / a b Œ . Hence the series converges absolutely for . T k k B  / he radius of convergence is . 3 œ / At , the series , since the B œ „ / diverges n-th term does not approach zero. This follows from the fact that lim 8 Ä _ 8 8 8x / 8 # 8 œ " . È 1 10. We have , with , for . Therefore . 0 B œ / 0 B œ / 8 œ "ß #ß â 0 ! œ " a b a b a b B 8 B 8 a b a b Hence the Taylor expansion about is B œ ! ! / œ B 8x B 8 œ ! _ 8 " . Applying the ratio test, lim lim 8 Ä _ 8 Ä _ 8" 8 k k k k a b k k 8xB " 8  " x B 8  " œ B œ ! . The radius of convergence is . 3 œ _ 11. We have , with and , for . Clearly, 0 B œ B 0 B œ " 0 B œ ! 8 œ #ß â a b a b a b w 8 a b 0 " œ " 0 " œ " a b a b and , with all other derivatives equal to . Hence the Taylor w zero expansion about is B œ " ! B œ "  B  " Þ a b 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 a b a b a b w ww # $ with , for . It follows that 0 B œ  " 8xÎ "  B 8   " 0 ! œ  " 8x a b a b 8 8 8 8 a b a b a b a b a b 8"
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—————————————————————————— —— CHAPTER 5.
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