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Cleveland State University Mathematics 182 Spring Semester 2018 Quiz 6 Series Name: Nine series are shown in the grid. Find the three in a row. Divergent series are O’s and convergent series are X’s. For the three series that form the three in a row, write whether the series converges or diverges, indicate what test you used to determine the convergence or divergence and show how the test applies. Write your solutions on the back of this sheet. 1
X O O O O X X X X 7. Converges using the ratio test. lim i →∞ a i +1 a i = lim i →∞ 2 / (5( i + 1))! 2 / (5 i )! = lim i →∞ (5 i )! (5( i + 1))! = lim i →∞ 1 (5 i + 5)(5 i + 4)(5 i + 3)(5 i + 2)(5 i + 1) = 0 < 1 The ratio is less than one so the series converges. 8. Converges using the ratio test. lim j →∞ a j +1 a j = lim j →∞ 4( j + 1) 5 / ( j + 1)! 4 j 5 /j ! = lim j →∞ ( j + 1) 5 j ! j 5 ( j + 1)! = lim j →∞ ( j + 1) 5 j 5 ( j + 1) = 0 < 1 The ratio is less than one so the series converges. Note that in the denomi- nator, there is a 6th power of j , in the numerator only a 5th power, so the limit converges to zero which is less than one. 9. Converges by comparison to a geometric series. - 1 4 m < cos m 4 m < 1 4 m since - 1 cos( m ) 1. 1 4 m and - 1 4 m are convergent geometric series with r = 1 / 4 < 1, so cos m 4 m must also be convergent. 2
Cleveland State University Mathematics 182 Spring Semester 2018 Quiz 6 Series Name: Nine series are shown in the grid. Find the three in a row. Divergent series are O’s and convergent series are X’s. For the three series that form the three