5

Find the Laurent Series of the function 2 +4 flz) = 22(22 + 32+2)where 1 < |z/ < 2...

Question

Find the Laurent Series of the function 2 +4 flz) = 22(22 + 32+2)where 1 < |z/ < 2

Find the Laurent Series of the function 2 +4 flz) = 22(22 + 32+2) where 1 < |z/ < 2



Answers

Find the Taylor series of each function at the indicated number. Give the interval of convergence for each series. $$f(x)=\ln (1+2 x) ; x=2$$

Okay, we are looking at Chapter 12 Section five, Problem number 17. We are asked to find the tailor Siri's expansion for the given function in this case, F of X equals two over one plus x squared and get the interval of convergence as well. So I'm going to start with an elementary Siri's that we know that resembles this as close as possible. So I can start with the one over one minus X, which we already know is the expansion of which I should say is one plus X plus X squared plus. And then we go till the end of term. It should be X to the end. And of course it keeps going. And this would be true over the interval for negative one is less than X is less than one. So I need to change my wonderful one minus sechs to two over one plus x squared. Going to do this in two steps on the start with the denominator, and I'm gonna replace Okay, the X with negative X squared so that when I rewrite my elementary, Siri's has won over one minus and now it's instead of ex, sweep it in the negative X squared. So to write the expansion, we're going to simply replace every X that we see with negative X squared. So the one is gonna stay one that then we're gonna have less negative X squared that then I've been replaced the X and the X squared. So it's gonna be quantity negative, X squared, squared. And we'll just keep going to the term which will be negative X squared to the so for the second steps. And now we're looking good. We have one over one plus x squared, but we needed to in the numerator so I can do that by multiplying term by term by two. So now I wind up with two over one plus X squared, which is the function that we were looking for and the expansion I'm going to simply take my previous expansion, multiply each term by too. So I'm gonna write that out as two times one and then plus two times the negative X squared and then two times the negative x squared, quantity squared and keep going to the end of term. I multiplied by two times naked of X squared to the end, and so I can write a little nicer my final expansion. As to then we have a minus two x squared and then a naked of squared. So we're back to class two X to the fourth, and then we would be at minus, uh, continuing the pattern that would be cubed, right, So it would be to x to the six plus. And then if we keep going to the end of term, you can save. Plus, we're alternating so I can pull that negative out of the power and just write as negative one to the end. Nice little term way of showing indicating back and forth between positive and negative. So then we still have the two than X. So powers and powers would multiply so extended to you and class and then let that done our interval of convergence. We're gonna start with our original interval of convergence from our elementary. Serious of negative one was an ex. Listen, one gets more room here, so starting with negative one less than X, less than one. Remember, our first step was to replace every ex with negative X squared. So I'm gonna do that here for my interval of convergence. So replacing X with negative X squared so I could get rid of the negative. Negative one turns negative. One is positive, but we switcher inequality X squared. Such a inequality. Negative one. And if I take the square roots? Uh, well, actually, I don't even need to do that with the X squared is still gonna be the same thing, right? So I'm still going to just feel the right. It s negative. One less than X, less than one. So our interval of convergence then didn't change. So we're still gonna be from negative one toe. Positive one. So there we go. I hope that helps.

Hello. We're looking at Chapter 12 Section five, Problem number 21. And we're to try to find the tailor. Seriously. Expansion for the given function f of X equals natural log of one plus two x to the fourth and give the interval of conversions. So I'm gonna start with the elementary. Seriously, Taylor Siri's expansion that most resembles this, which would be our basic natural bug expansion, which is the natural log of one plus X. And that expansion starts with the first term X. And then I have minus X squared over two. Um, and then plus X cued, get three minus. So we have an alternating Siris. And if we go to the term, we can write that a snake it of one to the end to account for the alternating. Then we have X to the end over and And this is true for the interval of negative one less than X, less than one. So we need to change this. Um, this is a pretty quick change. We can simply replaced this X with two X to the fourth and one shop. So we're gonna write that we're going to replace every X with two x to the fourth. So that will wind up with the expansion for the natural log of one plus two X to the fourth. And so we're gonna shoot through our elementary, Siri's, and everywhere we seen X, we're gonna replace it with two X to the fourth. So starting with that original X for the first term, so say two extra fourth groups, minus some might. Wow, I should think that to me. So minus We have what to X to the fourth quantity squared over two and then back to class. So to extent of worth. And this time we're cubed over three, minus a man. Clean it up a little bit. This is gonna be some Team X to the fourth. And then we have, um so this would be we could write. It is two squared X to the fourth over, too. I'm rather than simple. Find that will help us write the term nicer so I can write it. Us two squared and then extra the four square this X to the aid over to recognizing that Yes, to sweet cancel here, right would just have to extricate um, now and then plus too cute over X to the 12. I'm not over. Sorry. Times X to the 12 over three. Then we have minus stuck duct that so are empty term. I'm going to start with negative one to eat to accommodate for the alternating science. And then we have to to the and plus one power Now, then axe so technically and plus one power. But then times for right because of the two extra the force. So we were right that as X to the distributing before I could neither write four times and plus one or four. And plus for all that is over our denominator and plus one. And for the interval of convergence, we're simply going to replace our ex with our rich from our original interval Convergence with are two x to the fourth. So did that over here. So I'm starting at negative one less than thanks less than one. I'm replacing that X went to X to the fourth. And so if I divide everything by to get negative 1/2 less than thanks to the fourth, less than positive one have. And if I take the fourth route of the twos and the denominators I can say then that are interval of convergence. It's not allowing me to write down there. My interval of convergence is negative. One over the forth bridge of two to positive one over the fourth street. Up to there we go. So I hope that was simple.

We can rewrite this function as FX. He calls one over miners. One miners, ex miners too. Which yukos miners. One miner's miners, ex miners too. Over one And here you echoes miners ex miners too. So we can use this formula no need equals miners. Uh huh. From 0 to Infinity. My nurse one power and and X -2 to the power of you. You're cheeky girls assam miners one and plus one. And Arabs miners too. And next the convergent interval. No chance Miners as miners too should be greater than -1. All smaller than one. Richmond's. Yes. My nurse too. Greater than -1 or smaller than what and it has That's greater than one, smaller than straight.

Mhm We can rewrite this. Equating us X -2 times log two plus ex miners too. I'm going for this truck. It is just log two times one of us A half at -2. Where she calls Block two Plus Log one plus this term. And then if we let U equals a half times X -2. Then for this term we can use this formula An 80 girls log too. Class. The sub and because one to infinity admires two to the power. Oh yeah, so sorry it should be add Smyers 2/2 the power of enhance this equation. Oh this expression cause it's minus two times block two plus this song and design. He calls Log two times X -2 glass the Sun and then adds miners too to the power of and platform. And next we want to find the interval convergence. Now it is half times he acts miners too. Greater than -1 is smaller than equal to one. So X -2. Greater than -2 is smaller Now equal to two, so X is greater than zero. Must monitor the echo 24 That is explains to the interval 0- four.


Similar Solved Questions

5 answers
Mathematics 301 005 Spring 201814 .sequence is defined recursively as follows: 12an for all n 2 0 01 = 1 an+2 =Tan+l a0 = 0 and for the general terT 07 ' representation) Find explicit formula (a functional an
Mathematics 301 005 Spring 2018 14 . sequence is defined recursively as follows: 12an for all n 2 0 01 = 1 an+2 =Tan+l a0 = 0 and for the general terT 07 ' representation) Find explicit formula (a functional an...
5 answers
4. aJUse a double integral to find the total area enclosed by rose r = 4 sin(30)-b) Compute the area inside circle of radius and outside the ros0 given in problem 4.
4. aJUse a double integral to find the total area enclosed by rose r = 4 sin(30)- b) Compute the area inside circle of radius and outside the ros0 given in problem 4....
5 answers
Find the general solution of the differential equation. (Enter your solution as an equation.) Sy In(x) Xy' =0,* > 0
Find the general solution of the differential equation. (Enter your solution as an equation.) Sy In(x) Xy' =0,* > 0...
5 answers
Example 27.12akDetermine the magnelic dipole momont = electron orbitta pclon hydrogem aiom; assuming Ino olectron goundi slale und that Ciriuin T orbod wah _ FedJe 0.520210Part AAonembu UualIhe magnotc dpoa momonteNIA (and hero Ne1)ValueUnitsSubmitRoquaalanawge
Example 27.12ak Determine the magnelic dipole momont = electron orbitta pclon hydrogem aiom; assuming Ino olectron goundi slale und that Ciriuin T orbod wah _ FedJe 0.520210 Part A Aonembu UualIhe magnotc dpoa momonte NIA (and hero Ne1) Value Units Submit Roquaalanawge...
5 answers
F o sin : dx
f o sin : dx...
5 answers
Fk) =3,6/x =25,35f (xt)3.60 .80 1.20 0.90 0.722.0 3.0 4,0 5.0
fk) =3,6/x =25,35 f (xt) 3.60 .80 1.20 0.90 0.72 2.0 3.0 4,0 5.0...
4 answers
Suppose - matrix4x5 and has pivot positions Docolumnsspan 3 Explain
Suppose - matrix 4x5 and has pivot positions Do columns span 3 Explain...
5 answers
The curve defined by $2 y^{2}-x^{3}-x^{2}=0$ is called a Tschirnhausen's cubic.a. Plot the curve using the viewing window $[-1.5,1.5] imes[-1.5,1.5]$b. Find the volume of the solid obtained by revolving the region enclosed by the loop of the curve about the $x$ -axis.
The curve defined by $2 y^{2}-x^{3}-x^{2}=0$ is called a Tschirnhausen's cubic. a. Plot the curve using the viewing window $[-1.5,1.5] \times[-1.5,1.5]$ b. Find the volume of the solid obtained by revolving the region enclosed by the loop of the curve about the $x$ -axis....
1 answers
Show that $y=\log (1+x)-\frac{2 x}{2+x}, x>-1$, is an increasing function of $x$ throughout its domain.
Show that $y=\log (1+x)-\frac{2 x}{2+x}, x>-1$, is an increasing function of $x$ throughout its domain....
5 answers
Label the features on the Neanderthal skull
Label the features on the Neanderthal skull...
5 answers
For the price function given by$$p(x)=800 /(x+3)-3$$find the number of units $x_{1}$ that makes the total revenue a maximum and state the maximum possible revenue. What is the marginal revenue when the optimum number of units, $x_{1},$ is sold?
For the price function given by $$p(x)=800 /(x+3)-3$$ find the number of units $x_{1}$ that makes the total revenue a maximum and state the maximum possible revenue. What is the marginal revenue when the optimum number of units, $x_{1},$ is sold?...
5 answers
3x+5 if X<1 Ix-1 X21For the functionf(x)Evaluate f(-1)Evaluate f(1)Evaluate ((0)Evaluate f(5)Graph the function
3x+5 if X<1 Ix-1 X21 For the function f(x) Evaluate f(-1) Evaluate f(1) Evaluate ((0) Evaluate f(5) Graph the function...
5 answers
What is a formula for the nth term of the given sequence?15,21,27._an = 15 6(n _ 1)an =21 + 6(n _ 1)Submit Answeran =216nOn =3+6(n + 1)
What is a formula for the nth term of the given sequence? 15,21,27._ an = 15 6(n _ 1) an =21 + 6(n _ 1) Submit Answer an =21 6n On =3+6(n + 1)...

-- 0.020644--