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NAMES: MATH 152 April 1, 2015 QUIZ 7 • Show all your work and indicate your final answer clearly. You will be graded not merely on the final answer, but also on the work leading up to it. 1. (3 points) Determine whether or not the series converges. Show support for your answers. ∞ X 1 2 n + n ln n n=1 Solution: This is done using the direct comparison: Note that n2 +n1 ln n < n12 . So ∞ X n=1 ∞ X 1 1 < <∞ n2 + n ln n n=1 n2 (by p-test) and the series converges. 2. (3 points) Determine whether or not the series converges. Show support for your answers. ∞ X 1 n ln n n=2 Solution: Using the integral test: ∞ Z ∞ dx = ln(ln x) = ∞ x ln x 2 2 gives that the series diverges. (let u = ln x) NAMES: MATH 152 April 1, 2015 3. (3 points) Determine whether or not the series converges. Show support for your answers. ∞ X 1 √ n ln n + n n=2 P 1 Solution: This is done using the limit comparison test with ∞ n=2 n ln n : 1 √ n ln n+ n lim 1 n→∞ n ln n Thus since P 1 n ln n n ln n √ n→∞ n ln n + n 1 √ = lim = 1 6= 0 n n→∞ 1 + n ln n = lim diverges (by 2), the limit comparison test gives that the series diverges.