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Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {n!}{e^{n^2}} $

converges

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Campbell University

Oregon State University

Harvey Mudd College

Boston College

So we're giving a Siri's and um to determine if it converges or diverges. It's going to be easiest to use the ratio test because we have a factorial and the power eso. What we want to do is take the limit of the sequence at a sub n plus one over a sub, Ben. So if that's ultimately gonna look like is something like the absolute value of and plus one factorial over E and plus one. Yeah, sorry, that's going to be times eat, do that and Baird over and factorial and we're taking when, as an approaches infinity of all this eso What we end up getting is that we can cancel some of this stuff out and we'll end up getting the limit as an approaches infinity of n plus one over e the two and plus one. This is just through some factoring and cancelations that we can simplify this So you take the limit as an approaches infinity. We see that the bottom is going to grow faster than the top eso The limit is going to be equal to zero, um, and sense of limits equal to zero. We see that, um, this converges and therefore are Siri's will converge is well

California Baptist University