Problem 12 Easy Difficulty

Test the series for convergence or divergence.
$ \displaystyle \sum_{n = 1}^{\infty} (-1)^{n+1} ne^{-n} $

Answer

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Video Transcript

Let's test the Siri's for convergence or diversions. Now the first thing that stands out to me is this negative one to the end, plus one power that tells us that we're dealing with an alternate Siri's. So we should probably try alternating Siri's test. This is a nice test we can use to see if a Siri's converges. So first we have to define our BM, and this is just a positive part of a N. So Anne is just this end term of this sequence, not including the sun and then being is you just take the absolute value. Just take the positive part. So here, if you want, this is your formula for being in general, and here we see that it's positive check. That's the first requirement, and that's basically just by definition. And now for her, too. We need that the limit of being zero. So in this case, let's rewrite This is Lim and goes to infinity and overeat of the end, and this limit will be zero. And if you don't see why immediately, you could just use Low. Patel's rule derivative of the numerator is just one. Where is the denominator? Will still be exponential, so the denominator will still go to infinity and one over Infinity is era, So that's the second condition. Now we have one more condition. We need that the sequence bien is decreasing, so there's two ways to do this One ways. They just go ahead and check manually by hand. Use this formula so you know this is equivalent to and plus one overeater. The M plus one is less than or equal to it and over each of them. That's one way you can check if this is true, and if this is true, then it's decreasing. Another way to do this. Is it just to find the function affects as Ex Overeating X and then we know that the derivative is negative is equivalent to saying that F is decreasing. But efforts decreasing is a global into saying that the sequence bien decreases. So all we have to do here is just take a derivative of this function. We could use the potion rule, and if it's eventually negative, then that'LL answer Part three here and we'LL get a We'LL have our final answer, so let's go to the next stage here toe compute that derivative. So here, if prime of ex, use the potion rule so that gives us either that x minus x either the x o ver either the two x that denominators positive. But in the numerator, I can pull out of E to the X, and I know that this thing will be negative if one minus X is less than zero. Or, in other words, if X is bigger than one. This tells me that bien is decreasing when and is bigger than one. So really, I want tend to be bigger than or equal to two, because then is taking on the natural numbers. So if you're bigger than one, that means you're equal to two or larger than to. So this fantasize the third condition, and we can finally conclude that the Siri's converges the Siri's con Vergis. Let's write this up and let's explain why, by the alternating Siri's test, and that's our final answer