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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)

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  1. amistre64
    • 5 years ago
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    Do we write it out like a polynomial?

  2. amistre64
    • 5 years ago
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    What is your understanding of a power series?

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

  5. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    good..good..... what is your gut telling you about this problem?

  7. anonymous
    • 5 years ago
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    take the derivative

  8. amistre64
    • 5 years ago
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    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
    • 5 years ago
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    umm I don't know. but it makes it into \[f'(x)= 5/((1-x)^2)\]

  10. amistre64
    • 5 years ago
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    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
    • 5 years ago
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    better yet, is there a problem that you already know how to do that you can step me through?

  12. anonymous
    • 5 years ago
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    yeah hold on.

  13. anonymous
    • 5 years ago
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    \[f(x)=arccot (x)\] so... the derivative of that: \[f'(x)=-1/(1+x^2)\]

  14. anonymous
    • 5 years ago
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    meaning the integral of f'(x)=f(x)

  15. amistre64
    • 5 years ago
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    usually denoted in the textbooks a F(x) :) but yeah....

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

  17. anonymous
    • 5 years ago
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    so the integral of the power series=the power series of f(x)

  18. anonymous
    • 5 years ago
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    +c

  19. amistre64
    • 5 years ago
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    so step one, you found the derivative of arccot(x) right? and worked with it?

  20. anonymous
    • 5 years ago
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    yup

  21. amistre64
    • 5 years ago
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    then step one here is to find the derivative :) your gut was good....

  22. anonymous
    • 5 years ago
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    but I dont know how to deal with the whole being squared instead of just x

  23. amistre64
    • 5 years ago
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    5/ (1-x)^2 would it be better in expanded form? x^2 -2x +1 ?

  24. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    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
    • 5 years ago
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    Solution To do this problem let’s notice that 1/ (1-x)^2 = (d/dx) 1/ (1-x)

  27. anonymous
    • 5 years ago
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    crap that means i take the derivative again? -.- thats a double integral.

  28. amistre64
    • 5 years ago
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    lol...hold on its using the d/dx form later on.. might be helpful

  29. anonymous
    • 5 years ago
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    ok

  30. amistre64
    • 5 years ago
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    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
    • 5 years ago
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    okay i think i see where this is going

  32. amistre64
    • 5 years ago
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    this might be quicker :)

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  33. anonymous
    • 5 years ago
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    haha that is :)

  34. amistre64
    • 5 years ago
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    it was years before I knew what the "PrtScn" button was for... it captures a screenshot that you can psate into "Paint"

  35. amistre64
    • 5 years ago
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    *paste that is

  36. anonymous
    • 5 years ago
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    yeah i just figured that out about a year ago myself

  37. amistre64
    • 5 years ago
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    any of that jargon help you out with this problem?

  38. anonymous
    • 5 years ago
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    i think so I let you know shortly if the answer i get is right.

  39. anonymous
    • 5 years ago
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    wrong answer but i think i understand the concept better.

  40. amistre64
    • 5 years ago
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    at least one of us does :) good luck

  41. anonymous
    • 5 years ago
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    thanks :)

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