quiz 5 - note on p 699 for more details 2(3 pts Show that...

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Math 1B: Calculus March 3, 2010 Quiz 5 Lecturer: Prof. Mina Aganagic GSI: Gary Sivek Name: Answers 1. (2 pts) Is ± n =2 1 n ln n convergent or divergent? Justify your answer. Since x and ln x are increasing functions of x ,itfo l lowsthat 1 x ln x is a decreasing function of x whenever x> 1, where it is also continuous and positive. Then by the integral test, it converges if and only if the following integral converges: ² 2 dx x ln x =l i m t →∞ ² t 2 dx x ln x =l im t →∞ ² ln t ln 2 du u ( u =ln x, du = dx/x ) Since lim t →∞ ln t = ,th islastin tegra liss imp ly ³ ln2 du/u 1 / 2
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Unformatted text preview: note on p. 699 for more details. 2. (3 pts) Show that the following series is divergent or Fnd the sum: ∞ ± n =2 1 n 2-1 Using the techniques of partial fractions (details are omitted here), we Fnd that 1 n 2-1 = 1 / 2 n-1-1 / 2 n +1 . Then for large enough k , this becomes a telescoping sum: s k = k ± n =2 1 n 2-1 = k ± n =2 ´ 1 / 2 n-1-1 / 2 n + 1 µ = k ± n =2 1 / 2 n-1-k ± n =2 1 / 2 n + 1 = k-1 ± n =1 1 / 2 n-k +1 ± n =3 1 / 2 n s k = 1 / 2 1 + 1 / 2 2-1 / 2 k-1 / 2 k + 1 = 3 4-1 / 2 k-1 / 2 k + 1 ±rom this we see that lim k →∞ s k = 3 / 4, so the sum converges to 3/4 ....
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