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8.3 IntegralTest - R e v ie w S e r ie s 1 H a r m o n ic s...

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Review Series Harmonic series: 1 n n = 1 diverges. Geometric: ar n 1 n = 1 or ar n n = 0 1. A geometric series converges to a 1 r if r < 1 . The series diverges if |r| ≥ 1. 2. Telescoping: a n n = 1 a n + c
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Tests 1. a n n = 1 converges to L if the sequence of its partial sums {S n } converges to L. lim n →∞ S n = L ( ) 2. The n th Term Test for Divergence a n n = 1 diverges if lim n →∞ a n fails to exist or is different from zero. ( If lim n →∞ a n = 0 , the test is inconclusive.) Ex. Let a n = 2. Does {a n } converge or diverge? Does a n n = 1 converge or diverge?
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Ex. Let a n = n 2 n . Does {a n } converge or diverge? Does a n n = 1 converge or diverge? Ex. Let S n = n 2 n . Does a n n = 1 converge or diverge?
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8.3 Integral Test Sometimes we can’t find the exact sum of a series, but we can determine whether it’s convergent or divergent. Ex. 1 n 2 n = 1 Pg. 523: A series a n n = 1 of nonnegative terms converges if and only if its partial sums are bounded from above. Out[40]//TableForm= n Sn 0. 0. 100. 1.63498 200. 1.63995
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