Unformatted text preview: so it’s generally a good idea to think about doing a comparison before doing the integral test. EXAMPLE 10.32 Does ∞ X n =2  sin n  n 2 converge? We can’t apply the integral test here, because the terms of this series are not decreasing. Just as in the previous example, however,  sin n  n 2 ≤ 1 n 2 , because  sin n  ≤ 1. Once again the partial sums are nondecreasing and bounded above by ∑ 1 /n 2 = L , so the new series converges. Like the integral test, the comparison test can be used to show both convergence and divergence. In the case of the integral test, a single calculation will conﬁrm whichever is the case. To use the comparison test we must ﬁrst have a good idea as to convergence or divergence and pick the sequence for comparison accordingly....
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 Fall '07
 JonathanRogawski
 Math, Calculus, Derivative, Sequences And Series, Tn

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