M 408D SERIES PROBLEMS DAVID BEN MCREYNOLDS Problem 1 Easy Medium Determine whether the following series converge or diverge X n2 3 n 1 X n 1 X n 1 X 3n n 1 X 1 2 n 1 1 sin X e n n nn 2n 1 sin n 5n n 1 X 1 n sin n 2 nn n 1 X 3n 1 n2 1 n2 ln n n 1 7n X n5 3n 1 n 1 X n 2 X 1 n n 2n 3 n 1 n 1 1 X X 3 tan n n n 3 n 1 n 1 1 n 1 X X e n n 1 Problem 2 Easy Medium Evaluate the following limits cos x lim x 2 ln sin x lim x 0 1 1 ex 1 x lim x r k 2 1 x k2 1 6x 3x ax b x lim 10 x lim lim x x 0 x 0 4x x sin x 12 x1 x sin x x lim lim lim ex e2x x 0 x x x x Date February 6 2002 1 2 DAVID BEN MCREYNOLDS Problem 3 Medium Determine whether the following series converge or diverge Some of these are ridiculously hard X n 2 X n 1 X 2n 1 1 ln n ln n nn nnn n 1 n 1 10n n2 n X 22n 4n 1 n 1 X cos 2n X n n nn n 1 X 1 ln 2 n n 1 X n3 3n2 3n 1 X 3n n n 1 6 n 1 n 1 n2n Problem 4 Hard Suppose that we have the following for functions f x and g x f 0 1 g 0 f 0 0 1 f 0 g 0 0 1 g 0 2 Then find lim x 0 g x f x Problem 5 Easy Determine whether the following series converge or diverge X n ln n 1 n 2 X n 1 X n 1 X 1 2n n3 ln n n 1 3n X 6n 1 1 n3 n 4n 10 n 3 2n n2 n 1 X n 1 ln n n2 X n nn n 1 X nn 1192n n 1 X n 1 1 n1 001 Problem 6 Medium Suppose that an is a sequence of numbers and lim nan L n where 0 L then what can you say about X n 1 an M 408D SERIES PROBLEMS 3 Problem 7 Easy Determine whether the following series converge or diverge X 1 n X X 2n 1 n 24n n ln n n 1 4n2 1 n 1 2n 1 n 2 X n7n X 2 n X 3 n n n n 3 2 n 1 n 1 n 1 n 2 X 10 n X X n n 1 ln n 1 n 3 3 n n 1 n 1 n 1 Problem 8 Easy Determine whether the following series converge or diverge X X 1 2 ne n ne n n 1 n 1 n 1 X 1 n X 1 1 n n n2 1 n 1 n 1 X X n 1 n 1 n n 3 X n 1 1 n n 1 X n 10 10n 1 n 1 X 2n 1 n 1 2n 1 Problem 9 Easy Medium Determine whether the following series converge or diverge X n 2 1 n ln n ln ln n 4 DAVID BEN MCREYNOLDS Problem 10 Easy Medium Determine whether the following series converges conditionally absolutely or diverges X 1 n X 1 n n2 1 n 1 X n 1 X n 1 X n 1 X 4n4 1 n n 1 X 1 n 2n 3 3 2n n 1 n 1 1 e X 1 n 3 n n 1 n2n n 1 X 1 n cos n n 3n X 1 n 3n n 1 n 1 n n 2 n 1 X cos n 1 n n n X 1 n ln n n ln n n 2 3n2 5 n 1 X 1 n 3 4 1 n n3 8 X 1 n n 2 1 n sin n n 1 X 1 n ln n 2 n 1 n Problem 11 Limit Comparison Use the Limit Comparison Test to determine the convergence or divergence of the series X n 1 X n 1 X n 1 X n 2 n 1 n 3 n n 2 n 2 X n 1 X 1 n 6n3 n 1 X 1 n2 1 n 1 2n2 1 3n5 2n 1 X 1 n n2 1 2 n n3 1 X n 1 n 1 2n 1 5 n3 5n4 3 Problem 12 Alternating Series Test Use the Alternating Series test to determine the convergence or divergence of the series X 1 n 1 n 1 X n 1 X n 1 n 1 n 2n X 1 n n 2 X n 1 2 1 n 1 en e n ln n X 1 n n2 n 1 1 n 1 n n 2 X 2 1 n 1 n 1 en e n n2 1 2n 1 X 1 sin n 2 n 1 X 1 n 1 ln n 1 n 1 n 1 M 408D SERIES PROBLEMS 5 Problem 13 Ratio Test Use the Ratio Test to test for convergence or divergence of the series X n 3n n 1 X n 2 3n n 1 X n2 X 2n n 1 2n n5 n 1 X 4n n 1 X n 1 n 3n n 1 n Problem 14 Root Test Use the Root Test to test for convergence or divergence of the series X n X n n n X n n 2 n 1 n 1 2n 1 n 1 n 1 n 1 X ln n n X 2n n X 1 n n 1 n n 1 n 1 n 2 ln n Problem 15 Integral Test Use the Integral Test to test for convergence or divergence of the given series X n 1 X n 1 1 n 1 n e X n e n 1 cos n X n 1 X ne n n 1 nk 1 nk c X n 1 n2 n 3 6 DAVID BEN MCREYNOLDS Problem 16 Do the following series converge or diverge Why Show all work X X X n n2 n n n 1 X n 1 X n 2 X n 2 X n 1 n3 2n n2n 1 n n 1 n ln n n 1 nn n 1 n n 1 X n n n3 n n 1 X 2n 1 2n n ln n n n 1 X 1 n3 n n2 1 n 1 n n n 1 n 1 n 1 X n4 1 X n n 1 n n 1 X n2 n 1 X n 1 n 1 en n n 1 X n 1 1 n ln n X n n2n n 1 Problem 17 Compute the following limits Show work 1 n1 1 2n lim xln …
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