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Converting infinite series into an integral - intuition

  1. Jan 25, 2012 #1
    Question : How do i convert an infinite series into an integral?

    I searched a few sites and the method given is as follows
    replace r/n by x
    repalce1/n by dx
    replace Ʃ by ∫

    which works perfectly fine when i tried a few examples but i dont understand the intuition behind it. Why this method works? IS there any kind of derivation? Why this is been done and how it works - please explain me

    Thank you

    P.S. - i just joined physicsforum
     
  2. jcsd
  3. Jan 25, 2012 #2
    What level of math background do you have? Take a look at how integral is defined via Riemann sum and see if that makes sense.
     
  4. Jan 25, 2012 #3
    Didnt make much sense - my text only gave me the rules of converting an infinite series to an integral as stated above for which i am trying to find an intutiion. I am a 12th grade student by the way - so is it under my scope to get the derivation of converting an infinte series to an integral??

    IF not provide me the link or give the solution so that i am atleast satisfied that the method does not come out of the blue
     
  5. Jan 25, 2012 #4
    Would this link be clearer? If not, which part is not clear?
     
  6. Jan 25, 2012 #5
    oh i think you got my question wrong - your link tells me how we can find integration through summation too which is not what i am asking
    Ill give you an example of what i am trying to ask
    the question comes like this :-
    Find value of lim n--> ∞1/n Ʃ 1 + r/n ........limits of summation is from 0 to n
    So when i convert this to an integral i get the answer as ∫(1+x)dx limits :- 0 to 1

    Its easy to do this by following rules : replace r/n by x, 1/n by dx
    IF and b are limits of summation then limits of integral will be limit n --> infinity a/n and b/n

    These are the rules to convert infinite sum to an integral. But my question is : From where does these rules come from? Any derivation?


    A very general way is
    IF summation is lim n-->∞ 1/n Ʃ f ( a + b. r/n)
    then we can write it as an integral as ∫f(a + bx) dx
    limtis of integral can be found by the rules given above

    So i want to know how this happened. How do we convert series to an integral
     
  7. Jan 25, 2012 #6
    Hi jd!

    yenchin's link is quite good. As he says, it comes from the very definition. They key is just to understand the definition, from discrete sums to continuum (integral) in the limit. I think there is no general rule, no "mechanical" way.
    Go to the definition and set this sum as a Riemann sum from a partition of an interval like [0,1]. It could also be [1,2] while integrating the identity f(x)=x, so, no general rule.

    Get prepared to apply this when you study some physics. I began to understand mathematical analysis at the same time I began to study physics.
     
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