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JH
Numerade Educator

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Problem 36 Easy Difficulty

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {\sin(n \pi /6)}{1 + n\sqrt{n}} $

Answer

CONVERGES

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Video Transcript

Let's show that the following series is absolutely conversion. So to do this instead of looking at the original Siri's, we should look at the absolute value of the terms and then take the son and let me go ahead and rewrite that denominators and to the three over, too. Now we know that sign of any number in absolute value. It's no larger than what so I'LL go here and I'll just replace the new murder with the one and I'LL leave the denominator as it is so you can see him. I'm kind of setting myself up for comparison test due to the inequality that's due to this inequality over here and now this I can go ahead and obtain another upper bound. So let me just go ahead and remove the one in the denominator, and that will make the fraction larger because the new operators were the same. However, the denominator on the right is smaller, so smaller denominator means larger fraction. And now I know this Siri's over here, the Siri's conversions by the Pee test. It's a P Siri's with P equals three over to which is bigger than one, and that's when it converges. And now therefore, because of inequality. This means that our original Siri's no, by the comparison test our cirie, not theory, regional one, but the one that we had when we use absolute value. So by the comparison test this Siri's converges, and by definition it means that the original Siri's is absolutely commercial, So that's our final answer.