Question

Power Series: I'm having trouble understanding the concept. Here is one problem that I need help with:
Find the power series representation for f(x)=(4+x)/(1-x)

Answer 1

Do we write it out like a polynomial?

Answer 2

What is your understanding of a power series?

Answer 3

umm hardly anything at all really. I understand what we are trying to do (at least I think I do) but i dont know how to do it.

Answer 4

:) from the notes online it looks like it is a "sum" of something... what can you tell me about them.. in general

Answer 5

sums? that its a sequence with all the numbers being added together. so \[\sum_{n=1}^{\infty} 1/n\] would be (1+1/2+1/3....) and so forth.

Answer 6

good..good..... what is your gut telling you about this problem?

Answer 7

take the derivative

Answer 8

I have almost zero concepts about this so your driving.... tell me, how does the derivative help us? We can find the derivative quite easily, but what does that do for us?

Answer 9

umm I don't know. but it makes it into \[f'(x)= 5/((1-x)^2)\]

Answer 10

that is correct for the derivative :)

tell me...every time ive seen the big E symbol it was talking about "integration". Would that be any help for us here?

Answer 11

better yet, is there a problem that you already know how to do that you can step me through?

Answer 12

yeah hold on.

Answer 13

\[f(x)=arccot (x)\] so... the derivative of that: \[f'(x)=-1/(1+x^2)\]

Answer 14

meaning the integral of f'(x)=f(x)

Answer 15

usually denoted in the textbooks a F(x) :) but yeah....

Answer 16

and power series f'(x) of that is \[\sum_{n=0}^{\infty} (-1)^nx^(2n) \] (that's x raised to the 2n)

Answer 17

so the integral of the power series=the power series of f(x)

Answer 18

+c

Answer 19

so step one, you found the derivative of arccot(x) right? and worked with it?

Answer 20

yup

Answer 21

then step one here is to find the derivative :) your gut was good....

Answer 22

but I dont know how to deal with the whole being squared instead of just x

Answer 23

5/ (1-x)^2

would it be better in expanded form?

x^2 -2x +1 ?

Answer 24

no the standard form for the power series i'm dealing with right now is 1/(1-x) so i have to manipulate it into that form. The five is easy enough to get rid of by simply factoring it out but the (1-x)^2 is harder.

Answer 25

found this, might help if you understand this stuff :)

Example 4 Find a power series representation for the following function and determine its interval of convergence.

g(x) = 1/ (1-x)^2

Solution....

Answer 26

Solution

To do this problem let’s notice that

1/ (1-x)^2 = (d/dx) 1/ (1-x)

Answer 27

crap that means i take the derivative again? -.- thats a double integral.

Answer 28

lol...hold on its using the d/dx form later on.. might be helpful

Answer 29

ok

Answer 30

Then since we’ve got a power series representation for

1/(1-x)

all that we’ll need to do is differentiate that power series to get a power series representation for

Answer 31

okay i think i see where this is going

Answer 32

this might be quicker :)

Answer 33

haha that is :)

Answer 34

it was years before I knew what the "PrtScn" button was for... it captures a screenshot that you can psate into "Paint"

Answer 35

*paste that is

Answer 36

yeah i just figured that out about a year ago myself

Answer 37

any of that jargon help you out with this problem?

Answer 38

i think so I let you know shortly if the answer i get is right.

Answer 39

wrong answer but i think i understand the concept better.

Answer 40

at least one of us does :) good luck

Answer 41

thanks :)