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JH
Numerade Educator

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Problem 35 Medium Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{k = 1}^{\infty} (\sin 100)^k $

Answer

$$-0.336$$

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Video Transcript

let's determine whether the series converges or diverges. And then if it converges, we'll go ahead and find the sum. So we see that this is a series of the form like you have. Some number are and it's being raised to the K power. So this is geometric. And in our problem, we see that the are equal sign of 100. It's just a number, and we know geometric will converge only if we have the conditioned. Absolute value. Are is less than one. So really, we want to make sure that we know we know. First of all, that sign satisfies this. It's always between negative one and one, but it can sometimes equal negative one and one. But we would like to make sure that sign of 100 is not equal to plus or minus one. And the reason for this is so that the absolute value of our will not equal one. And the reason that we know that sign of 100 can't be plus or minus one is because of this fact. So let's come over here. If sine X equals one. This means X is equal to pi over two plus two Pi j. And this in a number of this form will never equal 100 because this over here is an irrational number. Whereas 100 or 100 over one, if you want, we can see that that's a fraction of two integers. So this is rational. Yeah. And similarly, if sign equals negative one, this means X is equal to negative pi over two plus two pi j. But once again, we have This can never equal 100 because this is also irrational. So we have that sign of 100. Yeah, is strictly less than one, and therefore this series will converge. Geometric with are less than one and then we'll go ahead and find the sum. That was the second question up here. So the sum let's write this sign 100 to the K. So we know the formula for geometric series. It's that first term, the one that you get by plugging in K equals one in this case, over one minus R. So we have sign of 100. This is after you get you plug in K equals one here and then one minus sign of 100 mm. And this will be the some of the series. So in summary, this given series up here gs geometric, it converges. And down here in the bottom, right, that's the some of the series, and that's our final answer.