Converting infinite series into an integral - intuition

[jd12345]

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 don't 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

 

[yenchin]

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.

 

[jd12345]

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 infinite 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

 

[yenchin]

Would this link be clearer? If not, which part is not clear?

 

[jd12345]

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

 

[Tarantinism]

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.