quiz 5.2 - note on p. 699 for more details. 2. (3 pts) Show...

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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 ) 2 convergent or divergent? Justify your answer. Since x and ln x are increasing functions of x ,itfo l lowsthat 1 x (ln x ) 2 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 ) 2 =l i m t →∞ ² t 2 dx x (ln x ) 2 =l im t →∞ ² ln t ln 2 du u 2 ( u =ln x, du = dx/x ) Since lim t →∞ ln t = ,th islastin tegra liss imp ly ³ ln2 du/u 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 ln n n + 1 Using our log rules, we Fnd a telescoping sum: s k = k n =2 ln n n + 1 = k n =2 (ln n-ln ( n + 1)) = k n =2 ln n-k n =2 ln ( n + 1) = k n =2 ln n-k +1 n =3 ln n s k = ln 2-ln ( k + 1) rom this we see that lim k s k =- , so the sum diverges . This examples shows why we have to be careful to deFne convergence as a limit of partial sums this way. Otherwise, if we just blindly cancel out terms in an inFnite sum, we might claim that the series converges to ln 2, which is positive, even though every summand is negative!...
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