WEBVTT
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Okay. The first step for finding a taylor series
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is find some derivatives. So we'll start with sine
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of X. And then it's first derivative coastline necks
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. Second derivative minus cynics, their derivative. When
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this goes my next fourth derivative back to syntax again
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then it will repeat. Okay now you have to
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find those function values at a so we're gonna plug
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in pi pi is zero F. Prime of Pie
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is-1. Double Prime Occupies zero. Triple Prime
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of Pie Positive one. Yeah. Next 10.
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And then they start over get-101 dot dot dot
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. Okay so now I'm going to write the taylor
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series. I'm gonna call T F. X.
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It's the constant zero plus the derivative times X minus
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A. Which is by to the first power over
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one factorial plus zero times X minus pi square over
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two factorial plus one. I'm on the third derivative
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now X minus pi cubed over three factorial plus zero
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x minus pi to the fourth over four factorial.
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Um Let's do one more minus one plus minus one
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plus minus one. X minus pi to the fifth
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over five factorial. Yeah. Okay so notice what
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we got is-1 X minus pi. It's the
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1/1 factorial plus one. X minus pi to the
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third over three factorial minus juan X minus pi to
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the fifth over five factorial. Okay if you can
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guess what the next one is going to be then
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you totally understand the series. So it should be
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add one x minus pi to the seventh over seven
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factorial. All right, so now we just have
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to write it in an enclosed form. So notice
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that it alternates. So we need a-1.
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And we're going to start this at I don't know
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what letter you use. I'm gonna start it at
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K equals zero. So I don't want this to
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be K. Because then the first term which is
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-1 here would be positive. So I need this
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to be K plus one. So when you plug
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in zero you get-1. When you plug in
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one you get positive one. When you plug in
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to get-1. Okay then all of them have
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an X-9 but not to the K. Power
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. It's to the odd powers 135 and seven.
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So to get odd numbers you can always use two
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K plus one or two K minus one or two
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K plus three. Or to k minus three.
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So you have to pick which one works the best
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. Okay, so when we plug in zero we
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want to get one. When we plug in one
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we want to get three. When we plug in
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to we want to get five. So two K
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plus one. And then since this and the exponent
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and the number on the bottom match up, we
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know that that's two K plus one factorial. Okay
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, so that's the taylor series. Um now let's
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find its interval or a radius of convergence. So
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remember you do that by taking the limit as in
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RK whatever K goes to infinity of a K plus
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one over a K And setting it less than one
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. No. Okay, so since this is an
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absolute value, we don't have to worry about the
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-1. So we're going to take the limit as
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K goes to infinity X minus pi to the two
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K plus one in parentheses. Plus one Over to
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K-plus one plus one factorial. And then we're going
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to divide it by the ak term which means we
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can just multiply by its reciprocal to k plus one
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factorial and then x minus pi to the two K
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plus one. All right, so this thing to
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K plus one plus one is two K plus 2
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+12 K plus three. We have the limit as
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K goes to infinity X plus one. To the
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two K plus three Over to K-plus three factorial.
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UK Plus one Factorial. An x minus pi to
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the two K plus one. Whoops. Okay,
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so let's look at that. I don't know why
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I put x plus one because I was being messy
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up above there. I think x minus pi.
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It's what I mean here, X minus pi.
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So let's look at those. We have x minus
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pi to the two K plus three Over X-9
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To the two K Plus one. So to simplify
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Since it's a fraction you subtract the exponents. So
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you get X-9 to the to power. Can
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we still have the limit here And it's by to
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the to oh and the absolute value is still here
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too. Yeah. Okay now let's look at the
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factorial is we have two K plus one factorial on
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the top and two K plus three factorial on the
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bottom. So let's just look at an example.
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So for example if Kay was three then the top
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is Seven factorial and the bottom is nine factorial.
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So what you're left with is two of them on
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the bottom the last two. So if K was
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a 100 Then it's 201 factorial on the top 203
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factorial on the bottom. So every time you're just
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left with two on the bottom it's the last one
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and the one right before it. So the last
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one is two K plus three. And so the
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one right before it is too K plus two.
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Mhm. All right. Now take the limit as
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K goes to infinity. Oh if k goes to
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infinity the bottoms infinity and the top is a number
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a number divided by infinity zero, zero is less
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than one. So this uh um it's convergent everywhere
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. So it's radius of convergence is infinity because we
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got zero when we took the limit and zero is
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less than one. That means convergent everywhere. Okay
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hope you loved it. I've ever given myself a
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smiling face