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Determine whether the series converges or diverges. Ifit converges, find its sum ( 32n-1...

Question

Determine whether the series converges or diverges. Ifit converges, find its sum ( 32n-1

Determine whether the series converges or diverges. Ifit converges, find its sum ( 32n-1



Answers

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {n!}{n^n} $

Don't let's determine whether the Siri's conversions or diverges first. Not that this is less than or equal to one over and square. Why? To see if this is true, Just multiply the denominators to the other side. And here, in order for this to be true, we should take and this ex for in here to be bigger than or equal to tool. So eventually if we take this sum here, if we're letting and be bigger than two, then this is smaller than this. And eventually Lee and his bigger than or equal to two because we started one and then we go all the way to infinity. So we passed two right away so I can go ahead and right this if you let's go and pull off that first term and then I have the sum from n equals two to infinity. The reason I'm doing this I'm pulling out the first term and rewriting the sum is because I would like to use comparison test. But if I want to compare our expression toe one over and square, I need end to be tour larger. And this is why I originally and started at one so I took that term out. Now I just have to to infinity so I can go ahead and replace this with one over and square and again this already using is this. This inequality is just using this fact here and then taking a sum on both sides. And then the plus one just came along for the ride here. Now I know that this Siri's here converges from section eleven point three. This is what we call P series, and P is bigger than one. In any time that happened. Opens. It's conversion. And if we just add one through a convergence Siri's, that's not going to change the fact that we still have a real number. So our Siri's is less than or equal to a conversion. Siri's Our Siri's is less than or equal to a conversion series. So by the comparison test, our Siri's also that Burgess Funny convergence, John, And that's our final answer

Let's determine whether the Siri's converges or diverges. So, first of all, I'LL claim that this is a This sum is less than or equal to the sum from one toe infinity of one over and minus one time's end. Now the reason for this is because, and factorial is equal to one times to all the way up Tio. And so this means that in factorial is larger than and minus one times in and Sense and factorial is larger. But over here we see it's in the denominator so the inequality will go in the other direction. So in other words, this fraction is larger because it's denominator smaller. That's what this show's over here. And then now we can use comparison test. But in order to do so, we should see that this Siri's comm urges. So for this one, there are many ways to go. You could try to use the Lim comparison test, so we're looking at the Siri's that's boxed in red. So let that let this term B an and then let it be in, just be one over and square. Then let's look at the limit of a N over beyond this is the limit comparison test. That's just and swear over and times and minus one. And let's evaluate this. You could use low Patel's rule here if you want. Instead, let me just go ahead and divide top and bottom bye and square. And then I get limit one over one, minus one of her head. Now let's go ahead and take that limit one minus one one over one minute zero, which is just one. And then we know that this Siri's will converge. That tells us that this Siri's one over and minus one times in converges. So this is using the limit comparison test. L see Teo abbreviate that now we can use the direct comparison test to explain why our Siri's circled on blue convergence. So, since first of all, we should point out, as the theory states, that we're dealing with a Siri's with positive terms so sense this one over and Factorial is always positive. We've shown that this Siri's converges by the direct computers and test. So this is not the limit comparison. This is the usual comparison. So we used both comparison test in this problem, but we only use the Lim a comparison to show that the larger Siri's convergence and once we realize that by the direct comparison that tells us that our original series converges and that's your final answer.

Let's determine whether the Siri's converges of averages well, either the one over end since one over end is always bigger than zero. Eat to the one of her end is bigger than E to the zero, which equals one. So here I can take our Siri's and say, This is not that way. I should do other inequality. This is bigger than or equal to the sum from one to infinity of just one over end sense E to the one over end. It's bigger than one, so we're just replacing the numerator eat to the one over and was something smaller, So the fraction is a hole gets smaller. And how about the Siri's here of one over end the Siri's diverges. This is known as the harmonic series, and the reason and diverges were Book proves it. But another way to prove it is just to use the pee test. And here P equals one that's the power of and in the denominator. And any time this number is less than or equal to one, you'LL will have divergence. So that's not for our Siri's. This is for the Green Series, the one over and squared Hope or the excuse me the one over and that someone there now to explain why our Siri's diverges since either the one over and over and his positive these are always bigger than zero. We've just shown that this Siri's the one in question, each of the one over and over and diverges bye, the computers and test. We just compared our Siri's with the smaller Siri's the lower bound, which would happen to be harmonic series that divers, so by comparison, are larger. Siri's and Red also has toe diverge, and that's your final answer.


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