quiz7ws2009 - -2 n n h). X n =1 n (ln n ) n Bonus: Around...

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Quiz 7 Instructor: Stephan Spencer 5 April, 2009 Student Name: 1. Use either the comparison test or the limit comparison test to determine whether the series converges or diverges. State which test you are using and fully support your answer. a). X n =1 3 n + n b). X n =1 n + 5 3 n 7 + n 2 c). X n =1 ± n 3 n + 1 ² n d). X n =1 n n - 1 e). X n =3 1 ln(ln n ) f). X n =1 1 2 n + 3 g). X n =1 1 n n 2 - 1 h). X n =1 5 n 3 - 3 n n 2 ( n - 2)( n 2 + 5) 2. Determine whether the following series converge or diverge. Make sure you state the test you use and verify all of its hypotheses. a). X n =1 ( - 1) n - 1 1 n b). X n =1 ( - 1) n - 1 3 n + 1 1 + 2 n 1
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c). X n =1 ( - 1) n - 1 ln( n ) n d). X n =1 ( - 1) n ± ln( n ) ln( n 2 ) ² n 3. Determine whether the following converge absolutely, conditionally, or diverge. State the test that you are going to use and justify your answers. a). X n =1 ( - 1) n n 4 b). X n =1 ( - 1) n - 1 n 2 2 n n ! c). X n =2 ± - 2 n n + 1 ² 5 n d). X n =1 n 10 10 n e). X n =1 n 2 e n f). X n =1 n n (2 n ) 2 g). X n =1 ± n
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Unformatted text preview: -2 n n h). X n =1 n (ln n ) n Bonus: Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula 1 = 2 2 9801 X n =0 (4 n )!(1103 + 26390 n ) ( n !) 4 396 4 n William Gosper used this series in 1985 to compute the rst 17 million digits of . (a) Verify, by hand, that the series is convergent. (b) How many correct decimal places of do you get if you use just the rst term of the series? What if you use two terms? (c) What is the smallest number of terms you need to use in order to guarantee that you get the rst 100 decimals of ? (Hint: For b and c it would be wise to use a computer program, say mathematica.) 2...
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quiz7ws2009 - -2 n n h). X n =1 n (ln n ) n Bonus: Around...

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