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)
Do we write it out like a polynomial?
What is your understanding of a power series?
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.
:) from the notes online it looks like it is a "sum" of something... what can you tell me about them.. in general
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.
good..good..... what is your gut telling you about this problem?
take the derivative
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?
umm I don't know. but it makes it into \[f'(x)= 5/((1-x)^2)\]
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?
better yet, is there a problem that you already know how to do that you can step me through?
yeah hold on.
\[f(x)=arccot (x)\] so... the derivative of that: \[f'(x)=-1/(1+x^2)\]
meaning the integral of f'(x)=f(x)
usually denoted in the textbooks a F(x) :) but yeah....
and power series f'(x) of that is \[\sum_{n=0}^{\infty} (-1)^nx^(2n) \] (that's x raised to the 2n)
so the integral of the power series=the power series of f(x)
+c
so step one, you found the derivative of arccot(x) right? and worked with it?
yup
then step one here is to find the derivative :) your gut was good....
but I dont know how to deal with the whole being squared instead of just x
5/ (1-x)^2 would it be better in expanded form? x^2 -2x +1 ?
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.
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....
Solution To do this problem let’s notice that 1/ (1-x)^2 = (d/dx) 1/ (1-x)
crap that means i take the derivative again? -.- thats a double integral.
lol...hold on its using the d/dx form later on.. might be helpful
ok
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
okay i think i see where this is going
this might be quicker :)
haha that is :)
it was years before I knew what the "PrtScn" button was for... it captures a screenshot that you can psate into "Paint"
*paste that is
yeah i just figured that out about a year ago myself
any of that jargon help you out with this problem?
i think so I let you know shortly if the answer i get is right.
wrong answer but i think i understand the concept better.
at least one of us does :) good luck
thanks :)
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