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JH
Numerade Educator

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Problem 24 Medium Difficulty

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

$ \displaystyle \sum_{n = 1}^{\infty} ( - 1 )^n \frac {2^n n!}{5 \cdot 8 \cdot 11 \cdot \space \cdot \cdot \cdot \space (3n + 2 )} $

Answer

absolutely convergent

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Video Transcript

let's use the ratio test to determine whether the Siri's converges are diversions. Now let me denote this by an then you could go ahead and right out the following for the ratio test. Now, in this case for the numerator, this will be a little more delicate. So we have to go ahead and replace all of the ends in the formula and the expression with n plus one. So here and the numerator will have to end plus one and then and plus one pants Auriol. Now, if you take a look at the denominator, this will turn into the product all the way up to now replace and within plus one. But this term here is just three in plus five. So that means that when the denominator will have five, eh? And we'LL also have three and plus two right before we have three and plus five. And the reason I know three and plus two will appear is because each because the terms air increasing by three each time and these two last two terms differ by three and we'LL see in a moment why that's important. Why, that's an important fact now in the denominator. We just have our usual and here. So just go ahead and put that back in. Now, as usual, when you're taking one fraction and dividing by another, you could just flip that fraction over the blue fraction and then multiply. And you could also see there's no need to write the negative one to the end power because we're taking absolute value and all the remaining terms air positive. So we can actually stop writing naps in value mar Now in plus one factorial, let me replace that with, in fact, for riel times and plus one. Let me justify that by definition and plus one factorial is just a product of the first and plus one numbers. And if you just group the first and together that's a product, they're multiplied. Not not Sorry. Looks like a subtraction, but that this should be multiplication here. And then we see that that first term is just in factorial. So that justifies this over here and then Denominator, I'll keep that as it is. And then now that we flipped the blue fraction upside down and now you can see why was important before till right out this three and plus two term. And the reason for now is because we can cancel this five eight all the way up to three and plus two we'll cancel nicely with the same terms in the denominator. Also, the reason for using this fact over here about the factorial Sze is that I could cancel this in factorial with this other in factorial and we have some more cancellation here. We could take off this tune to the end with this one up here and we're still left with a two on top. So we'll end up with Lim and goes to infinity two still haven't plus one there and on the bottom, all less love left over is three in plus five and at this point you could replace the ends with the X and use low Patel's rule if you want. In either case, you see that this limit is two over three, which is less than one. So we conclude by the ratio test that the Siri's convergence and that's our final answer