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4. Much harder than today’s problems 3. C

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7. Test the convergence or divergence of the following infinite series, indicating the tests used to arrive at your conclusion: (a) (3) (b) (4) (a) < 1.(M2) and hence the series converges by the ratio test. (R1) 3 5.

8. (c) (5)

9. Discuss the convergence or divergence of the following series: (a) (4) 6. (b), k is a positive integer. (5)

10. Determine whether the series converges. (Total 4 marks) = 1. (M1)(A1) This can be done using comparison with the harmonic series. (R1) Let b n = represent the harmonic series. Since b n diverges, so does a n. (A1) 4 7.

11. (a)Use the ratio test to calculate the radius of convergence of the power series (3) (b) Using your result from part (a), determine all points x where the power series given in (a) converges. (5) (Total 8 marks) (M1) (A1) (R1) (M1) (R1) (M1) (R1) 8.

12. Test the convergence or divergence of the following series (a) (5) (b) (5) (Total 10 marks) Note: Do not accept unjustified answers, even if correct. is a decreasing sequence in n (M1) (M1) (C1) = 0 (M1) so the series is convergent, by the alternating series test. (R1)5 9.

13. (a)Describe how the integral test is used to show that a series is convergent. Clearly state all the necessary conditions. (3) (b)Test the series for convergence. (5) (Total 8 marks) (R1)(R1) (R1) (M1) (A1) 10.

14. Find the range of values of x for which the following series is convergent. (Total 7 marks) The ratio test must be used.(M1) The series must converge if for n sufficiently large 11.

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