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JH
Numerade Educator

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Problem 8 Easy Difficulty

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?
$ \displaystyle \sum_{n = 1}^{\infty} \frac {(-1)^{n - 1}}{n!} $

Answer

converging to about 0.6321

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Video Transcript

let's calculate the first a terms of the sequence of partial sons. So four decimal places. So that's asking for us one as to all the way. There s a will have to write all of these out. Okay, Now we know by definition, that s n is just a one. That's the starting point all the way up to I am so as one is just a one. So this is just the term that you get when you plug in and equals one here. So they're Wanda, put just one. And then, for example, as to that's a one plus a two. So when we plug in one, we just have one. And then when you plug in and equals two minus a house so that's point five. And for the remaining values, if we like, we could even come over here Instead of adding fractions, we could just use Wolfram Alpha. So here is You could see this is the right code because we're able to compute these partial sums. We've already done the first two. Now, let me add the first reasons they want, plus a two plus a threes. You can see by the signal notation. This is the sun for history in this four at the very in lets me know that I just want four decimals. So here we have point six six six seven. So since we're just rounding off here, this may not be exact. SEL. We should use approximation to be precise, he er now adding the first four terms increase that arena floor point six two five zero. Now going to five terms. We're halfway there. Change that formula Five, but and compute that. And we have point six and then three threes now going to six terms. Change that five foot six and we have six three one nine now onto the seventh. We have six three, two one. And then finally, going to that eighth and final term will get point six three two one again. So that's that might tell us something there. So that that answers the first part of this question. We found the first eight terms round that off to four decimals. Then there's a second part of this question. Does it appear that the Siri's is conversion or diversion? Well, just by looking at the terms here, it does look like the limit exists. So here I would say conversion. And it looks like the limit of SN is about point six three two. And if you wanted a formal explanation for why the Siri's convergence, you could actually just use the alternating Siri's test and check that all the conditions are satisfied, and that's your final answer.