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JH
Numerade Educator

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Problem 21 Easy Difficulty

Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {(-0.8)^n}{n!} $

Answer

$\approx-0.5507$

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Video Transcript

Graff both a sequence of terms. That's a N and also the sequence of partial sums in the same screen. So that will be the essence. So here in Dismas and Purple, we have just the sequence. And so the first coordinate is telling you the value of end. So here I should maybe zoom out so we could see the first value. So here, where is my first values down here? I should have noticed that. So there you can see the entire purple graph. The first values negative point eight et tu is point three two approximation a three and so on. And then the red terms you could see in the bottom left Over here the red is the essence. These were the partial sums. So looks like the partial sums air starting to level off it about negative what half a little smaller than that About negative point five five. So that takes care of the first part. So I'm guessing it about not the best approximation That's just two decibels. Now that we'Ll have to use that they're in here tto find it correct afford a small places. So if we want the sun currents of four decimal places. This means that we want there, which is less than or equal to bien plus one. Now I'LL explain what I mean Here the bien is just point eight to the end over and factorial It's just coming from over here ignoring the negative part. This is alternating Siri's, of course. So this is why I'm using the altar mate, kneading Siri's estimation there. I'm here to get the upper bound. So this is one of using in terms to approximate the entire song by s I mean the entire incident. Some once infinity. Yeah. So now we like to know what this number is less than zero point zero zero zero zero five. The whole point of that number is that just and sure that were correct that the four decimal places we don't want a number larger than this because if it is, it'll this five will. If we round off, it's going to make a one in the fourth spot. So now we have to solve this for end. So we have point eight and over in factorial less than this number over here and using a calculator. This is true as long as Anna seven or more, however, really, in this estimation there and we should be using and plus one so we want and plus one to be bigger than or equal to seven. But this means we want him to be bigger than or equal to six. Excuse me, I got really sloppy here. That's a seven up there and I'll explain in one second. So the difference here is the SN, which is the number of terms versus the bien plus one. So here we found and to be bigger than seven. But that's coming from the B n plus one. So really we wanted and plus one the sub script on the B to B more than seven. That's equivalent and Big six or more. This means that we just want to use six terms in the sun. So if we want to be correct to six decimals, so this means that we could just approximately his own the entire sum. So let's write that out. It's just approximately as six. That's four decimals. If we write this up negative point five five o six, that's correct before decimals, and that's our final answer

JH
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