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Pre-Calc Homework Solutions 393

# Pre-Calc Homework Solutions 393 - Chapter 9 Review an an(n...

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6. lim n ) } a a n 1 n 1 } ) 5 lim n } ( n ( n 1 1 2 1 ) ) x ) ) ) x 3 ) n 3 1 n 3 } 5 ) x ) 3 The series converges absolutely for ) x ) 3 , 1, or 2 1 , x , 1. When ) x ) \$ 1, the series diverges by the n th Term Test. (a) 1 (b) ( 2 1, 1) (c) ( 2 1, 1) (d) None 7. lim n ) } a a n 1 n 1 } ) 5 lim n } ( n ( 1 2 n 2 1 ) ) 2 x 3) 1 2 n 1 1 ) 1 n 1 1 } ? } ( n ( 1 2 n 1 1 ) ) 2 x 1) 1 2 n 1 ) n } 5 } ) 2 x 2 1 1 ) } The series converges absolutely for } ) 2 x 2 1 1 ) } , 1, or 2 } 3 2 } , x , } 1 2 } . When } ) 2 x 2 1 1 ) } \$ 1, the series diverges by the n th-Term Test. 8. lim n ) } a a n 1 n 1 } ) 5 lim n } ( n 1 ) x ) n 1 1 ) 1 n 1 1 } ? } ) n x ) n n } 5 ) x ) lim n } ( n 1 1) n ( n n 1 1) n } 5 ) x ) lim n 5 } ) x e ) } lim n } n 1 1 1 } 5 0 The series converges absolutely for all x . Another way to see that the series must converge is to observe that for n \$ 2 x , we have ) } n x n n } ) # 1 } 1 2 } 2 n , so the terms are (eventually) bounded by the terms of a convergent geometric series. A third way to solve this exercise is to use the n th Root Test (see Exercises 57–58 in Section 9.5).
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