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).
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