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Use the Ratio Test to determine whether the series is convergent or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {2 \cdot 4 \cdot 6 \cdot \space \cdot \cdot \cdot \space \cdot (2n)}{n!} $

divergent

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Let's use the ratio test determined whether the Siri's converges. Now we should focus on this term right here. This is our man A N equals and the numerator. We have the product of the first and even numbers. So two, four, six all the way up to to end And then here we have in factorial. So for the ratio test, we look at the limit and goes to infinity and plus foreign over a n in the absolute value. And here we could drop the absolute values because an and and plus one are positive. So for a n plus one, let's do this and read in the numerator. So here we have two times, four times six all the way up to to end and then the last one when we increase and buy one, that's two and plus two the very last term. And then in the denominator we see this. But now we'LL have been plus one because of the plus one here that's the numerator now in the denominator and we already know what this is where he wrote the formula for and up here. So just go ahead and plug that in now Let's go ahead and write this out over here. So here we should go ahead and rewrite this as a product of fractions. So bring that in factorial into the numerator. And then I also let me rewrite this and plus one factorial, as in factorial times and plus one. And then we still have this product of the first and evens. Now we should cancel his munch. Can we see the end? Factorial We'LL cancel with this one. This is why we rewrote it here the two, four, six all the way up to two and cancels in the numerator as well and we're left over with the women is and goes to infinity. We have to end plus two over in plus one Oops and plus one. However, this limit is equal to two which is bigger than one. So going on to the next page toe right are summary or conclusion So the Siri's for months to infinity Write an infinity of here on the side two, four, six all the way up to to end over and factorial this dime urges again because the to the limit that we got was bigger than one by the ratio test. And that's our final answer