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JH
Numerade Educator

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Problem 5 Easy Difficulty

Determine whether the series is absolutely convergent or conditionally convergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {\sin n}{n^2} $

Answer

Absolutely Convergent

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Video Transcript

let's go ahead and show that this Siri's is absolutely conversion. So to do that well we should be looking at is not the original Siri's, but the Siri's that you get after you take the absolute value of the terms that your adding. So this here is here and that I just wrote, If this is a convergence series, then the original serious conversions. So if this Siri's converges, then the original is absolutely conversion. Therefore, let's look at this. Siri's here. Now we have sign and over and square. Well, you know, Sign is always less than or equal to one an absolute value. So let's just go ahead and replace sign with one. So here we no sign of end, less than or equal to one that justifies this inequality over here. So we're using the comparison test Now. If we look at this, Siri's here. This Siri's convergence by the PT Test. So it's a Pee series with P equals two that's bigger than one. So it emerges. So by comparison, test Okay, this Siri's due to this inequality over here. That's why we're using comparison. This tells us that if we look at the Siri's of Signe and over and square absolute value from one to infinity that this also converges now going on to the next page. Therefore, we've just shown that the original Siri's from one to infinity sign in over and square that this is absolutely conversion, and that's our final answer.