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Approximate the sum of the series correct to four decimal places.

$ \displaystyle \sum_{n = 1}^{\infty} ( - 1)^n ne^{-2n} $

$-0.1050$

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Oregon State University

Harvey Mudd College

Idaho State University

Boston College

Let's approximate the Siri's correct up to four decimal places. So first of all, we see that this is alternating. So here one would use the alternating Siri's estimation Kerem. But before this is used, what we would have to show that the serious convergence. So the Siri's we can show that a conversions by the alternating Siri's serum. So here, to do that, we can define B end to just be the positive part of the A. M. So just this part right here. These terms are both positive, so this is clearly positive, the limit being equal zero. This is true, and you can use low Patel's rule here to show this, and we can also show that the be ends are decreasing. If you want to show this, you can just define ffx to be ex e negative to X, and then show that the door evidence is negative, so that means it's decreasing. So by the alternating Siri's test alternating serious narrow, this series will converge. And now, by the alternating Siri's estimation there on this tells us that we can have a bound on the ear. This is less than our people to be in plus one. So the ear is when we approximate the entire Siri's, but only using in terms. Now this is by the serum. Moreover, if we want correct to four decimal places, we shouldn't make the air. This law, I'LL put four zeros in a five. So if we have any number of less than this, that means that there is not large enough to affect the fourth decimal spot, because when you round off, it will not change the zero into a one because the fifth decimal will be less than five. So this is why I used this inequality at the end. So using this and up here, we'd like to know when and e to the negative to end is less than this. So go on to the next page, using a calculator that turns out to be when N is bigger than six. But really, we wanted be an plus one not being so really we want and plus one bigger than or equal to six, and I gives us and to be five or more. So this means that we'Ll approximate using five terms. In other words, as if we just do as five. There's some music just of the first five terms, so I could even write that out. This is just the sum and equals one toe. Five negative, one to the end and e to the minus to end. Now we're on ly going toe five places. Here are four places. Excuse me. So I get negative zero point one zero five zero and that's correct. Up to four decimals, and that's our final answer.