0.2. Real analysis 5

Then the power series is convergent for |z| R, and is divergent for |z| R. For

|z| R a power series may be differentiated term-by-term, and the differentiated

power series has the same radius of convergence R.

To define what it means to say that a function is Riemann–integrable on an

interval [a, b], we start with a partition π, which is to say a sequence {xn} such

that

a = x0 x1 · · · xJ = b,

and choose interspersing numbers ξj so that

(0.7) a = x0 ≤ ξ1 ≤ x1 ≤ ξ2 ≤ x2 ≤ · · · ≤ ξJ−1 ≤ xJ−1 ≤ ξJ ≤ xJ = b.

A Riemann sum for

b

a

f(x) dx is then a sum of the form

S(π, ξ) =

J

j=1

f(ξj)(xj − xj−1) .

The mesh of π is defined to be

(0.8) mesh(π) = max

1≤j≤J

(xj − xj−1) .

That is, the mesh is the length of the longest subinterval defined by π. We say that

the integral exists and has the value I if for every ε 0 there is a δ 0 such that

if mesh(π) δ, then |S(π, ξ) − I| ε for any choice of the interspersing points ξ.

As to sums and integrals such as

a≤n≤b

un and

b

a

f(x) dx,

we have two different conventions. In a sum, we sum over all n that satisfy the

indicated constraints. Thus if b a, then there is no such n, and the value of

the sum is 0. However, for integrals, if b a, we simply say that the value of the

integral is −

a

b

f(x) dx.

The arc length of a parameterized curve (x(t), y(t)) for a ≤ t ≤ b is the supre-

mum of all sums of the form

(0.9)

J

j=1

(x(tj) − x(tj−1))2 + (y(tj) − y(tj−1))2

where a = t0 t1 · · · tJ = b.

Theorem 0.13. (Fundamental Theorem of Calculus, First Form) Suppose that

f(x) is Riemann-integrable on the interval [a, b]. For a ≤ x ≤ b, put

F (x) =

x

a

f(u) du .

Then F (x) is continuous on the interval [a, b]. If a c b and if f(x) is continuous

at x = c, then F (x) is differentiable at x = c, and F (c) = f(c).