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JH
Numerade Educator

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Problem 16 Medium Difficulty

Test the series for convergence or divergence.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {n \cos n\pi}{2^n} $

Answer

Converges

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Video Transcript

test the Siri's for conversions or diversions. Now the first thing we observe here is that it's actually alternating. It might not be obvious right away, but it's actually it's just due to this coastline term because if you were just right, this Siri's out. So the first one you plug in and equals one and then the next term so co sign is just bouncing around between pie into I And that just means that we're getting either one remind us one depending on then. So there's three pie, which is the same thing as co sign a pie again. So we could kind of see the repetition here. So this is just negative. One over two plus two over two square, minus three over to Cube and so on. So, yes, it is alternating. So here we should just take B end for if we want to use the alternating Siri's test, we should define beyond to just be the absolute value. And so this is just an over to to the end, because when we take the absolute value of co sign well, we noticed from our example for writing this out above Don't the co sign is equal to either plus or minus one. So this is just absolute value of plus or minus one, and that's just one. So that's why the one is just and write it out here. So the first condition fall training, Siri says, is that you're bien terms or positive. This is Troop, the second condition is that the limit of being ghost zero. This is also true, and one way to see this. It's just use love. It's a house rule, okay? And there's one more condition to check that the sequence being is decreasing. So me running out of room here will be Write this down here. So one more condition. So let's go out and check this. That's still the question we have to show this. So bien was an over to the end. So the question is whether the Siri's is decreasing so this we can go ahead and show this by because this is equivalent to just and plus one over two of the n plus one less than or equal to and over to the end, you can go ahead and check if this is true. Another option here is to justify enough in terms of B by just replacing and with X. And then we know that f decreases. That's acquittal into the derivative being negative time. So here lets his computer derivative of X using the question rule and then the denominator is always positive. But that numerator, the question is, is whether that's negative. So here this thing will be negative if one minus X, Ellen too, is negative. So that just means one over Ellen, too. Less than X. So if and is bigger than one over natural other too. Then BN is decreasing. Therefore, all three conditions for the alternately Siri's tests or satisfied. So therefore, by alternating Siri's test, the Siri's converges and that's your final answer.