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Calculues Questions

  1. Feb 12, 2010 #1
    1. The problem statement, all variables and given/known data

    A. Let [tex] g(x)= \sigma\frac{1}{sqrt(n)}(x^{2n}-x^{2n+1}) [/tex]
    Prove g(x) is continous in [0,1].

    B. Let f be a function such as f(0)=1 and there's a neighouhood of x=0 in which :
    [tex] f ' (x)= 1+(f(x))^{10} [/tex] .
    Find the MacLorin Polynom of degree 3 of f(x).

    2. Relevant equations
    3. The attempt at a solution

    Have no idea about it....

    Thanks in advance
     
  2. jcsd
  3. Feb 12, 2010 #2

    Mark44

    Staff: Mentor

    For A you need to show that g is continuous at each point in [0, 1]. What's the definition of continuity of a function at a point? Do you have to do this by using the definition of continuity or can you use the fact that this is a polynomial and all polynomials are everywhere continuous?

    For B you need f(0), f'(0), f''(0), and f'''(0). You already are given that f(0) = 1, and you have f'(x), which you can evaluate at x = 0.

    To find f''(0), you need to find f''(x), which you can do by differentiating f'(x), and then evaluate f''(x) at x = 0.
    To find f'''(0), differentiate f''(x), and then evaluate at x = 0.
     
  4. Feb 12, 2010 #3

    LCKurtz

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    That's "Maclaurin Polynomial".

    Well, don't you just need f(0), f'(0), f''(0) and f''(0) to calculate that series? You are given formulas for f(0) and f'(x). You must need a couple more derivatives. You might need the chain rule. Show us your derivatives.
     
  5. Feb 13, 2010 #4
    Well, B is completely understandable...
    About A->I need to show it's continous using power-series theorems...If I'll prove that the given power-series is convergeing uniformly to g - I'll be done...I've no idea about it...
    I'll be delighted to get some help

    Thanks!
     
  6. Feb 13, 2010 #5

    HallsofIvy

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    Power series? g(x) is a polynomial! You don't need to worry about any power series or convergence! A is not asking about a limit as n goes to infinity is it? It is just about a single polynomial for a fixed value of n.

    Or was that [itex]\sigma[/itex] supposed to be [itex]\Sigma[/itex]? That is, is this a sum over all n? In that case, because it is a power series, it converges uniformly inside its radius of convergence. You need only show that its radius of convergence includes [0, 1].
     
    Last edited: Feb 13, 2010
  7. Feb 13, 2010 #6
    That's excatly what I can't understand....how can I find the eadius of con. in this specific case?


    tnx
     
  8. Feb 13, 2010 #7

    Mark44

    Staff: Mentor

    TheForumLord,
    A fair amount of time has been wasted because we didn't understand what you were trying to communicate. It has now come to light that your first problem problem is a summation. The Greek alphabet has upper case letters and lower case letters. In particular, upper case sigma, [itex]\Sigma[/itex], is used to represent summations. Lower case sigma, [itex]\sigma[/itex], is used in statistics to represent the population standard deviation. I interpreted [itex]\sigma[/itex] in this problem as a constant. It didn't occur to me that you really meant a summation.

    Also, at this stage of your mathematical education, you really ought to learn how to spell "calculus." It's clear to me that you're not likely to be in the finals of a math spelling bee, but at least get calculus right.
     
  9. Feb 13, 2010 #8
    Dear Mark44... My english is pretty lame indeed but in this particular case, writing calculus in a wrong way was just a typing mistake - which can occure to anyone....

    I didn't know how to write Upper case sigma in Latex so plz don't judge me...
     
  10. Feb 13, 2010 #9

    Mark44

    Staff: Mentor

    Sure, anyone can make a typo, but you can eliminate at least some of them by looking over what you've written before you hit the submit button.
    [ tex] \sigma[/tex] or [ itex] \sigma[/itex] (without the leading space) produces [itex]\sigma[/itex].
    [ tex] \Sigma[/tex] or [ itex] \Sigma[/itex] (without the leading space) produces [itex]\Sigma[/itex].

    Same for all the rest of the Greek letters.
     
  11. Feb 13, 2010 #10
  12. Feb 14, 2010 #11

    HallsofIvy

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    Better is \sum :
    [tex]\sum[/tex].

    By the way, just clicking on a formula in any post will show you the LaTex code used for it.

    Your series can be written by separating the even and odd powers- it is [itex]\sum a_mx^m[/itex] with
    [tex]a_m= \sqrt{\frac{2}{m}}[/tex]
    if m is even and
    [tex]a_m= \sqrt{\frac{2}{m-1}}[/tex]
    if m is odd.
    As for finding the radius of convergence, using the ratio test gives
    [tex]|x|< \sqrt{\frac{m+1}{m}}[/tex]
    if n= 2m and
    [tex]|x|< \sqrt{\frac{m}{m-1}}[/tex]
    if n= 2m+1

    What is the limit of those as n goes to infinity?

    Of course, you will need to check if the sum converges at x= 1 but that is easy.
     
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