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Solving Integrals using summations

  1. Nov 5, 2014 #1
    1. The problem statement, all variables and given/known data
    Many places I have seen when solving integrals you change a lot of it into sums.

    http://math.stackexchange.com/quest...dfrac-tan-x1m2-tan2x-mathrmdx/1006076#1006076

    Is just an example.

    So in general, how do you solve integrals (CLOSED FORM) by using series?

    Thanks!

    (for example integrate [itex]\displaystyle \int_{0}^{4\pi} \sin(x) dx[/itex] using SERIES)?

    Thanks!

    2. Relevant equations

    [itex]\sin(x) [/itex]

    3. The attempt at a solution

    I am a beginner, with no expertise in this area, so I asked the question. I don't know any methods to start with. Any help will be appreciated.
     
    Last edited by a moderator: Nov 5, 2014
  2. jcsd
  3. Nov 5, 2014 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Do you know what the MacLaurin series (Taylor series centered on x= 0) is for sin(x)?
     
  4. Nov 5, 2014 #3

    FeDeX_LaTeX

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    Gold Member

    Tonelli's theorem says that if [itex]f_{n}(x) \geq 0 \text{ } \forall n, x[/itex], then we can interchange the sum and the integral as follows:

    [tex]\sum \int f_{n}(x) dx = \int \sum f_{n}(x) dx.[/tex]

    For general [itex]f_n,[/itex] Fubini's theorem says that if either [itex]\int \sum |f_n|[/itex] or [itex]\sum \int |f_n|[/itex] converge, then [itex]\int \sum f_n = \sum \int f_n.[/itex]
     
    Last edited: Nov 5, 2014
  5. Nov 5, 2014 #4

    Mark44

    Staff: Mentor

    Amad27,
    When you write a LaTeX expression, please use the correct tags. For itex, the closing tag is /itex, not \itex (in brackets). I fixed your post here and the LaTeX in your other post about limits.

    You can check that what you have done is correctly formatted by clicking the PREVIEW button.
     
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