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In summary, in physics, a sum can be changed to an integral when the indexing value becomes continuous and the term of the sum is comparable to the sum. An integral is essentially the limit of a sum with an arbitrarily large number of arbitrarily small elements. However, if one or more terms of the sum are too large, it cannot be considered as an integral. This can be seen when using sums to approximate the area under a curve, where a smaller delta x and a more continuous x sequence leads to the use of an integral. In Bose-Einstein condensation, the number of particles in the ground state must be separated from the integral because it becomes very large and must be accounted for separately. Further discussions about this topic should be made in

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

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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.

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closing this thread now.

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