# When we can change a sum to an integral?

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fxdung
In physics we often change a sum to an integral.But I am not clear when can we change a sum to an integral?When a term of sum is comparable to the sum,can we change the sum to integral?

Mentor
When the indexing value i of the sum becomes continuous like x .

Homework Helper
An integral is essentially the limit of a sum an arbitrarily large number of arbitrarily small elements.

AM

fxdung
Then if one(or more) term of sum is very great(can not consider as a small element) so we can not consider as an integral?

Mentor
Think about how you initially use sums to approximate the area under a curve.

you might compute the area as the sum of box areas where the width of the box is one and the length of the box is f(x) and use the sum of boxes with
x=1, 2, 3, 4, 5...

then to get a better approximation you use a width of one half for the box and the sequence with x=0.5, 1, 1.5, 2, 2.5 ...

so as delta x (aka box width and x difference) gets smaller and the x sequence becomes more continuous then an integral comes into play.

fxdung
I do not understand why in Bose-Einstein condensation, below critical temprature we must separate the number of particles of ground state(E=0) from the integral?Is that because the number of particles in ground state become very large so that we must separate it from the integral(the total number of particles)?

Mentor
You should ask that question in a separate thread with a related title As now you’re talking more physics and not math.