Lesson 24a - Absolute Conditional Convergence


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Unformatted text preview: n 1 en en This means that the absolute series diverges by the divergent 0 test | (1) n | n n2 n2 n as the limit does not go to 0. This also means the orignal series diverges as well as the limit does not exist (it shots up and down to positive or negative infinity due to the (‐1)n. Example 3 Determine if the following seriesconverges absolutely, conditionally, or diverges: (1) n n2 First we test for absolute convergence: 1 n ln(n) 1 1 | ( 1) | n ln(n) n 2 n ln(n) n2 n Now we test the new series, we notice that we can integrate this, so means we test the integral test conditions: 1. The terms are positive 2. The terms are clearly decreasing 3. The function is continuous (no breaks in the domain provided n>1) 1 1 This means we can integrate: dn du ln(u ) ln(ln(n)) n ln(n) 2 u 2 2 2 This means the absolute series diverges, so we check the conditional convergence. Since the original series is clearly alternating, decreasing, and limit0, we have that this series converges by the alternating series test. Thus this series is conditionally convergent....
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