# Converting infinite series into an integral - intuition

1. Jan 25, 2012

### 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 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. Jan 25, 2012

### 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.

3. Jan 25, 2012

### 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 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

4. Jan 25, 2012

### yenchin

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

5. Jan 25, 2012

### 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

6. Jan 25, 2012

### 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.