Uniform Convergent
Uniform Convergence Explained
Sequence of functions 
, 
, 2, 3, ... is said to be uniformly convergent to 
for a set 
of values of 
if, for each 
, an integer 
can be found such that
for 
and all 
.A series 
converges uniformly on 
if the sequence 
of partial sums defined by
converges uniformly on 
.To test for uniform convergence, use Abel's uniform convergence test or the Weierstrass M-test. If individual terms 
of a uniformly converging series are continuous, then the following conditions are satisfied.1. The series sum
is continuous.2. The series may be integrated term by term
For example, a power series 
is uniformly convergent on any closed and bounded subset inside its circle of convergence.3. The situation is more complicated for differentiation since uniform convergence of 
does not tell anything about convergence of 
. Suppose that 
converges for some 
, that each 
is differentiable on 
, and that 
converges uniformly on 
. Then 
converges uniformly on 
to a function 
, and for each 
,
, , 2, 3, ... is said to be uniformly convergent to for a set of values of if, for each , an integer can be found such that
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and all .A series converges uniformly on if the sequence of partial sums defined by
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.To test for uniform convergence, use Abel's uniform convergence test or the Weierstrass M-test. If individual terms of a uniformly converging series are continuous, then the following conditions are satisfied.1. The series sum
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is uniformly convergent on any closed and bounded subset inside its circle of convergence.3. The situation is more complicated for differentiation since uniform convergence of does not tell anything about convergence of . Suppose that converges for some , that each is differentiable on , and that converges uniformly on . Then converges uniformly on to a function , and for each ,
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