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Pre-Calc Homework Solutions 385

# Pre-Calc Homework Solutions 385 - Section 9.4 40 This is a...

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40. This is a geometric series with first term a 5 1 and common ratio } sin 2 x } . Since ) } sin 2 x } ) , 1 for all x , the interval of convergence is 2‘ , x , ‘ . Sum 5 } 1 2 a r } 5 5 } 2 2 2 sin x } 41. Almost, but the Ratio Test won’t determine whether there is convergence or divergence at the endpoints of the interval. 42. (a) For k # N , it’s obvious that a 1 1 1 a k # a 1 1 1 a N 1 n 5 N 1 1 c n . For all k . N , a 1 1 1 a k 5 a 1 1 1 a N 1 a N 1 1 1 1 a k # a 1 1 1 a N 1 c N 1 1 1 1 c k # a 1 1 1 a N 1 n 5 N 1 1 c n (b) Since all of the a n are nonnegative, the partial sums of the series form a nondecreasing sequence of real numbers. Part (a) shows that the sequence is bounded above, so it must converge to a limit. 43. (a) For k # N , it’s obvious that d 1 1 1 d k # d 1 1 1 d N 1 n 5 N 1 1 a n . For all k . N , d 1 1 1 d k 5 d 1 1 1 d N 1 d N 1 1 1 1 d k # d 1 1 1 d N 1 a N 1 1 1 1 a k # d 1 1 1 d N 1 n 5 N 1 1 a n (b) If a n
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