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Test the series for convergence or divergence.

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

divergent

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just has this Siri's for conversions are diversions, although the Siri's is alternating, That doesn't necessarily mean we should go straight to the alternating theories test. Because here, if we call this being or in this case, let's not name this. Let's just look at this term here in the limit. I claimed that this limit is non zero. So first of all, if we just write this out as a fraction now, the numerator is just end end times. Whereas in the denominator, that's one times two times three all the way up. Ten. Now, if we look at each of these fractions, these fractions are all bigger than or equal to one. So that means that this entire product is bigger than or equal to just and over one. So this line may equals infinities. So, in other words, this lemon is bigger than or equal to the limit as n goes to infinity of end just by using this inequality over here and that's infinity. So what that tells me is that as and it's large a M, which is negative one to the end, this number will Oska lee between between large, larger and larger alternating values. So here I mean alternating inside. So this means that the limit of a N is not able to zero because the limit just it does not exist. So the Siri's diverges by the diversions test, and that's our final answer. So the Siri's I didn't write the word I shouldn't die emerges by. The divergence is, and that's the final answer.