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Infinite Sum Theorem - Proof

For example,

3 Δx + 5 Δx2 ≈ 3 Δx - Δx2 + Δx3       (compared to Δx)

but 

3 Δx + 5 Δx2 ≈ 2 Δx (compared to Δx).

The Infinite Sum Theorem is used when we have a quantity B(u, w) depending on two variables u < w in [a, b], and the total value B(a, b) is the sum of infinitesimal pieces

ΔB = B(x, x + Δx).

he theorem gives a method of expressing B(a, b) as a definite integral.

 

Let B(u, w) be a real function of two variables that has the Addition Property in the interval [a, b] — i.e.,

B(u, w) = B(u, v) + B(v, w) for u < v < w in [a, b].

Suppose h(x) is a real function continuous on [a, b] and for any infinitesimal subinterval [x, x + Δx] of [a, b],

ΔB ≈ h(x) Δx (compared to Δx).

Then B(a, ft) is equal to the integral

06_applications_of_the_integral-4.gif

Intuitively, the theorem says that if each infinitely small piece AB is infinitely close to h(x) Δx compared to Δx, then the sum B(a, b) of all these pieces is infinitely close to 06_applications_of_the_integral-5.gif (Figure 6.1.2). This is why we call it the Infinite Sum Theorem.

06_applications_of_the_integral-6.gif

Figure 6.1.2

PROOF
Divide the interval [a, b] into subintervals of infinitesimal length Δx. Because B(u, w) has the Addition Property, the sum of all the ΔB's is B(a, b). Now let c be any positive real number. For each infinitesimal subinterval [x, x + Δx] we have

06_applications_of_the_integral-7.gif

Adding up,

06_applications_of_the_integral-8.gif

Now take standard parts,

06_applications_of_the_integral-9.gif

or

06_applications_of_the_integral-10.gif

Since this holds for all positive real c, it follows that

06_applications_of_the_integral-11.gif


Last Update: 2010-11-25