## anonymous 5 years ago 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)

1. amistre64

Do we write it out like a polynomial?

2. amistre64

What is your understanding of a power series?

3. anonymous

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.

4. amistre64

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

5. anonymous

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.

6. amistre64

7. anonymous

take the derivative

8. amistre64

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?

9. anonymous

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

10. amistre64

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?

11. amistre64

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

12. anonymous

yeah hold on.

13. anonymous

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

14. anonymous

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

15. amistre64

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

16. anonymous

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

17. anonymous

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

18. anonymous

+c

19. amistre64

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

20. anonymous

yup

21. amistre64

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

22. anonymous

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

23. amistre64

5/ (1-x)^2 would it be better in expanded form? x^2 -2x +1 ?

24. anonymous

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.

25. amistre64

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....

26. amistre64

Solution To do this problem let’s notice that 1/ (1-x)^2 = (d/dx) 1/ (1-x)

27. anonymous

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

28. amistre64

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

29. anonymous

ok

30. amistre64

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

31. anonymous

okay i think i see where this is going

32. amistre64

this might be quicker :)

33. anonymous

haha that is :)

34. amistre64

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

35. amistre64

*paste that is

36. anonymous

yeah i just figured that out about a year ago myself

37. amistre64

38. anonymous

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

39. anonymous

wrong answer but i think i understand the concept better.

40. amistre64

at least one of us does :) good luck

41. anonymous

thanks :)