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Determine whether the series converges or diverges.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n!} $

converges

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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.