WEBVTT
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Okay, So for part eight year, let's go
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ahead and use s ten to approximate or estimate the
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infinite some s And also we'LL also ask How good
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is the estimate? And so here by as ten
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of course we mean the end of the ten partial
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some. So the solution for part A So first
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will write out as ten. By definition, this
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is the sum of s just for the sum from
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one to ten. So in that case, just
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go ahead and write out the few terms here and
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add up a bunch of fractions. This would take
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quite a while here so one could go to the
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calculator, Miss round off here it's a one point
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five four nine seven six eight and that's that's an
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approximation to us. Now the answer, The second
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question, How good is the approximation? Well,
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if you look at exercise thirty four you'll see that
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they give the value for us for this sum here
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the infinite sum given by Oiler. So in our
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case, the error is the difference between the exact
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value in the approximation. So since we now have
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the exact value a plug that in and we'LL also
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play guitar approximation as ten, and in this case
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we're getting about point zero nine five two, which
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is less than one over ten. So it's not
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a bad approximation because it's less than one over ten
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, but we can do better by taking larger.
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And so it's not a bad estimation, considering we
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only use ten terms, but one over ten just
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might not be accurate up, depending on one's purposes
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. So let's go on to part B now for
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part B. This is world will actually go ahead
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and improve the estimate from party, eh? Using
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Inequalities three and plugging in and equals ten into three
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. So what's good and solve this so solution?
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So first, let's recall from party what we found
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. We found that s ten was about Okay,
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so now if you will get inequalities three it gives
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up or lower bounds through the exact value in terms
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of sn and these inner girls here. So notice
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the difference in the lower bounds of the intern girls
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ones and plus one the other one's the end.
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So here, because we're to use any equals ten
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, so that determines that. And if you recall
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from our Siri's and has won over and swear.
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And so that means FX should just be one of
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the explorer. So let's go ahead and plug all
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this in here. Okay? So then now we
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could actually just go ahead and integrate this we do
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have improper and liberals here, but these ones are
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not too bad, because when you plug an infinity
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, the expression become zero. So that would become
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s ten plus one over eleven, less than or
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equal to s less than or equal to as ten
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plus one over's him. So then now one would
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use the information from part, eh? And plug
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that in for the S tens here. So plug
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those in for as ten and then add the fraction
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and so going on to the next page. All
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right, this out. So there's s ten plus
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one over eleven and then s ten plus one over
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ten and then just add those together. And there
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we go now to find value for us here.
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So think of it this way. We have an
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interval here, a lower and upper bound. And
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all we know is that as you fall somewhere in
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between here. So instead of choosing either then points
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another approximation would just take the midpoint of these two
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the average and so here will take us to just
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be the average. Here. So is an approximation
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to s. And this becomes also we'd like to
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know the ear involved here, and so there's a
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few ways to do this. So this is our
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approximation for us and so sensitive in the middle,
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the air is just they have the length of the
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interval. And similarly, you can do this half
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if you like, but because we chose the mid
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point these two halves of the same length. So
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that's one way to write it. Or you could
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just take the entire honorable length and divided by two
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. So the entire in ruling it would be the
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right most end point minus the left most end point
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and divided by two. So this is Yeah,
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this is the length of the interval here. And
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if you go to the Congo later, you see
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that this is indeed less than the point zero zero
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five. So the ear is improved. Bye.
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Using in part B by using the midpoint s Now
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, let's go on to the next cage for party
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. This is where we'Ll compare the estimate in part
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B with the exact value that we mentioned in part
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A. But this is a coming for number thirty
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four. So this is in number exercise number thirty
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four. So solution. So from party, we
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had a approximation improved estimate for us. So let
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me write that down five to two to and we
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also saw what the ear bound was. So if
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you look at number thirty four as we mentioned,
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this gives the exact value dudes Oiler. So in
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this case, we see that the air given by
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using this value of s go to the calculator here
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zero zero zero to a and that's even that's also
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Weston the point zero zero five. So this shows
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that the answer we're getting from part B is getting
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very, very close to the exact answer here.
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Okay, so we have one more party here party
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. This will be the final part. This is
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where we'Ll find a value and you just need one
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value that'll ensure not the ear and the approximation is
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less than point zero zero one. Okay, so
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solution Well, we don't We don't want to use
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an equals ten anymore because that's not a good enough
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approximation. So instead, we're just we won't plug
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in a value for any it. But we'LL use
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this inequality for the air here. So for this
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is for the remainder when using in terms. So
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this will be one over X squared. That's R
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FX. All right, so it's pointing at that's
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remind us where the one over exports coming from.
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And we want this to be less than point zero
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one. This is what we want. And so
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we're gonna solve end to make sure we get what
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we want. So let's just go ahead and evaluate
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that in a girl there that'LL be a negative one
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over X from the power rule. And then if
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you plug in infinity, you get zero. If
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you plug in and you get one over end,
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simplify this and then we get an is bigger than
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one over point zero zero one, which is a
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thousand so any value of and larger than a thousand
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would work. So the smallest one that you could
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possible use is a thousand won any larger and would
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work, but here all they asked for is a
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value. So in that case, we could just
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use this value here and by what we just showed
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taking this value of and will ensure that the exact
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answer minus this value here, the one thousand and
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first partial. Some will indeed me less than point
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zero zero one, and so that resolves the problem
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.