How can you approximate a sum by an integral without having a small Δx?

[aaaa202]

The attached pdf shows integral approximations of two sums, which are done in my book. In the first there is no result but the book simply states that one can approximate the sum by an integral.
My question is: How is this done? Normally when you approximate a sum by an integral you have a sum of the form:
∑f(x)Δx, where Δx is small. But this is not the case here.

 

[vanceEE]

When you use an integral to approximate the sum, you would use a partial sum as an approximation and then use an integral with the term of the partial sum as it's lower limit and infinity as it's upper limit, assuming that you're dealing w/ a convergent series your integral will converge to the maximum error/remainder of your approximation.
Ex.

Let S₇ = 35.7
Then, $$ 35.7 ≤ S ≤ 35.7 + \int_7^∞ f(x)\ dx $$

Where $\int_7^∞ f(x)\ dx$ Is the maximum error.

 

[Ray Vickson]

The attached pdf shows integral approximations of two sums, which are done in my book. In the first there is no result but the book simply states that one can approximate the sum by an integral.
My question is: How is this done? Normally when you approximate a sum by an integral you have a sum of the form:
∑f(x)Δx, where Δx is small. But this is not the case here.

The exact sum has the form S = ∑f(x)Δx with Δx = 1. If f(x) changes without very much 'curvature' over the x-region of interest, there will not be much difference between the sum S and the integral I = ∫ f(x) dx.

Look at it this way: the graph of y = f(n), n=1,2,...,N has a "staircase" appearance, while the curve y = f(x), 1 ≤ x ≤ N is smooth and weaves its way between the discrete points (n,f(n). In fact, if f(x) is a straight line, the integral and the sum would be exactly equal because for each i the area of the rectangle with base 1 and height f(i) woul be the same as that of the quadrilateral with base 1 and sides of heights f(i-0.5) and f(i+0.5). If the graph f(x) curves slightly, there will be slight differences between the two. So, in the cited paper, the graph of f(n) curves slightly on 1 ≤ n ≤ N, provided that n* is large compared with N, and that is why the sum is approximately equal to the integral.