## 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

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 (Figure 6.1.2). This is why we call it the Infinite Sum Theorem.

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