Theorem 1. Suppose fn is a sequence of function defined on set A and suppose
that
∀n ∈ N and ∀x ∈ A |fn (x)| < an ,
P∞
P∞
then n=1 fn converges uniformly if n=1 an converges.
Definition 1. Let an be a sequence of numbers. Then expresion
∞
X
an (x − a)n
n=1
is called power series with a center at a.
P∞
Theorem 2. Let n=1 an (x − a)n be a power series. Then there exists R ≤ ∞
such that the power series converges if |x − a| < R and diverges if |x − a| > R.
The number R is called the radius of convergence and (−R, R) the interval of
convergence.
P∞
n
Theorem 3. Let
n=1 an (x) be a power series. If either of the following
conditions is fulfilled:
1. limn→∞ | aan+1
|=g
n
p
2. limn→∞ n |an | = g
then the radius R of convergence for that power series is equal to R =
g ∈ R+ , R = ∞ when g = 0 and R = 0 when g = ∞.
1
g
when
Theorem 4. Power series converges uniformly in the interval of cenvergence.
Definition 2. Taylor series for function f in point x0 is a power series given
by formula
f (x) = f (x0 )+f 0 (x0 )(x−x0 )+
f 000 (x0 )
f n (x0 )
f 00 (x0 )
(x−x0 )2 +
(x−x0 )3 +...+
(x−x0 )n +...
2!
3!
n!
1