Assessment 5.pdf - Weekly Assesment 5 Joshua Egger MATH-104...

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Weekly Assesment 5 Joshua Egger MATH-104 Prof. Iliopoulou Exercise 1: (i) If | a k | → 0, then + k =1 a k converges absolutely. (ii) If a k > 0 and 0 < a k +1 a k < 1 for all k N , then
This is false. Consider the sequence a k = 1 k as a counterexample. 0 < a k +1 a k = k k +1 < 1, but the harmonic series + k =1 1 k diverges. (iii) If a k 0, then + k =1 a k converges.
(v) If + k =1 a k converges, then + k =1 a k k converges.
Exercise 2: Test for convergence the following series: (i) + k =1 ( - 1) k k p , p R This series converges by the Dirichlet criterion for all p R + , since the sequence of partial sums of the series ( - 1) k is bounded by 1 for all k N and 1 k p is a decreasing sequence which converges to zero in the limit as k → ∞ . But, if p < 0 we have an oscillating sequence that does not converge, a k = ( - 1) k · k r where r > 0, so the preliminary test gives divergence in that case. (ii) + k =1 ( 1 2 ) 1 /k This series diverges by the preliminary test, since a k = ( 1 2 ) 1 /k 1 6 = 0. ( x 1 /k 1 for any x R
as was shown on assignment 3).

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