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JH
Numerade Educator

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Problem 19 Easy Difficulty

Use the Ratio Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {n^{100} 100^n}{n!} $

Answer

Absolute convergent

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Video Transcript

let's use the ratio test to determine whether this converges. So the ratio test requires that we look at this term and that's end to the one hundred power times, one hundred to the end power or ran factorial. So for the ratio tests, we will at the limit as n goes to infinity, absolute value and plus one over and here we could ignore the absolute value because all the numbers and our fractions are positive. Now let me do this numerator and red. So this will be and plus one to the one hundred, one hundred's and and plus one and plus one factorial. Now let's do the denominator in a different color. Still this and blew a n. We just use this formula appear to write that in and to the one hundred one hundred to the end and factorial. Now let's go ahead and cancel on as much as we can. But first, let's just rewrite. This is a product, so I have n plus one to the one hundred one hundred to the n plus one, and factorial comes up there on the denominator. I'LL have this term and let me rewrite this as in factorial times and plus one so that we could do some cancellation. We'LL also have into the one hundred and one hundred to the end, but cancel on as much as we can. We see that the in fact Orioles cancel, and if we look at the one hundred's, we could cancel out and of those and we're left with one over and we can also cross off one of the M plus ones. However, we still have this end to the one hundred and the denominator. And if we look at and plus one to the ninety nine over and to the one hundred, we can be right, this's and plus one over end to the ninety nine times one over end. And this term over here is basically one, but we're most applying it to one over end, and that's gonna go to zero in the limit. So this limits less than one. So we conclude that the Siri's want to infinity and to the one hundred, one hundred and over, and factorial converges coming by the ratios ist and that's our final answer