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JH
Numerade Educator

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Problem 18 Easy Difficulty

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

Answer

By the ratio test, the series is absolutely convergent.

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Video Transcript

let's use the ratio test to determine whether the Siri's conversions. So here, let's look at this term and a n equals and factorial over end to the in power. Now the ratio test. I suggest that we look at this limit here absolute value and plus one over. And so let's do the numerator and red So that will be and plus one factorial over end, plus one and plus one. Then that's being divided by AM A N is and factorial over and stand. That's right, the absolute value. You could ignore the absolute value because all the other fractions here consist of positive numbers. So now let's go ahead and simplify this. So let me rewrite this. And plus one factorial is and factorial times and plus one. Then I also have this and to thee and power Then I'LL have this term here. So let me write this as and plus one to the end times and plus one and we still have one term here and factorial. Now it's good and cancel on as much as we can. We see those factorial is go away and plus one terms cancel So we have the limit and goes to infinity and to the end, over and plus one to the end. Now I'm running out of room here. Let me take this to the next page. Now that's free, right, This's end over and plus one to the end. Now, this looks familiar. If you consider this fraction over here, the reciprocal split up that fraction and was even put it to the end power. If you recall this expression here, if you take the limit, this is equal to the number E. So this is the reciprocal of our terms. So that just means that our limit just goes toe one over here. Now this number's less than one. So we can conclude that the original Siri's the sum from one to infinity and factorial over into, then conversions by the ratio test. And that's your final answer.