Section10.3a-ConvergenceOfSeriesWithPositiveTerms

# 1 1 dx x2 an 1 n2 convergessoconverges n 1 a3 a2 1 n2

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Unformatted text preview: converges, 1 then so does the series. 1 1 dx x2 an 1 n2 converges, so converges. n 1 a3 a2 1 n2 a4 … n What’s the Difference? 1 n2 decreases faster than an 1 n2 1 n an 1 n Example 1 For what values of p does the series, , n converge? n 1 1 1 dx xp 1 xp is continuous, positive, and decreasing, if p>0. The integral converges if p>1. Diverges otherwise. Therefore, 1 np n 1 Converges for p>1 by Integral Test. Diverges for 0 < p ≤ 1 by Integral Test. Diverges for p ≤ 0 by Test for Divergence. p P‐ Series Test 1 np n 1 Converges for p > 1. Diverges otherwise. Note As with all series, convergence depends only on behavior of the “tail”. Examples Determine if the following series converge or diverge. 1 n4 n 3 1 n 4 n 1 n 1 1 n 1 4 Try It Show whether the following series converge or diverge. Pay careful attention to notation. Justify all steps. ne n n 1 1 n ln n n 2 en n 1 3n n2 n2 5 1 n2 5...
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## This note was uploaded on 02/24/2014 for the course APMA 1110 taught by Professor Morris during the Fall '11 term at UVA.

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