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# Use the Ratio Test to determine whether the series is convergent or divergent.$\displaystyle \sum_{n = 1}^{\infty} \frac {\cos (n \pi /3)}{n!}$

## absolutely convergent

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to use the ratio test to see if this converges. So we look at the limit n goes to infinity and plus one over and we're an is given by this term over here. So first we have the numerator and plus one so that'LL be co signed And then we have an plus one here and we're dividing that by the denominator here and and now we'LL go ahead and flip this blue fraction and multiply it to the green. And when we do that, let's see here. So we'll have co signed and plus one over three over another co sign and then in factorial and then and plus one factorial. So let's simplify be separately. So here, notice that and plus one with the factorial that can be written as and so would go ahead and cancel those factorial is there And also if you look at the's terms right here, so co sign pi over three. So, for example, if an is one just right out of a few terms of the sequence and you'LL see, it's it's periodic. We have a one af negative one half negative one negative one has one half and one and that would keep repeating. So that means that when if we divide consecutive terms here, well, if you divide this divided by that, you get one an absolute value. This divided by that would be to an absolute value. And let's see, here we have. And that looks like a largest fraction that we would get. It will be too. So this is less than or equal to the limit and goes to infinity two. And then we still have our n plus one from this term over here so again, with the absolute value is always going to be one or two. And so we'LL just use upper bound and replace this with the two that goes zero on the limit, which is less than one. So we conclude by this that the Siri's convergence by the ratio test

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